5 analytic trigonometry - verona public schools€¦ · 350 chapter 5 analytic trigonometry...

349

Analytic Trigonometry

5.1 Using Fundamental Identities

5.2 Verifying Trigonometric Identities

5.3 Solving Trigonometric Equations

5.4 Sum and Difference Formulas

5.5 Multiple-Angle and Product-to-Sum Formulas

5

11

14

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350 Chapter 5 Analytic Trigonometry

IntroductionIn Chapter 4, you studied the basic definitions, properties, graphs, and applications ofthe individual trigonometric functions. In this chapter, you will learn how to use thefundamental identities to do the following.

1. Evaluate trigonometric functions.

2. Simplify trigonometric expressions.

3. Develop additional trigonometric identities.

4. Solve trigonometric equations.

Pythagorean identities are sometimes used in radical form such as

or

where the sign depends on the choice of u.

tan u � ±�sec2 u � 1

sin u � ±�1 � cos2 u

5.1 Using Fundamental Identities

What you should learn● Recognize and write the

fundamental trigonometric

identities.

● Use the fundamental trigono-

metric identities to evaluate

trigonometric functions, simplify

trigonometric expressions,

and rewrite trigonometric

expressions.

Why you should learn itThe fundamental trigonometric

identities can be used to simplify

trigonometric expressions. For

instance, in Exercise 111 on page

356, you can use trigonometric

identities to simplify an expression

for the coefficient of friction.

Fundamental Trigonometric Identities

Reciprocal Identities

Quotient Identities

Pythagorean Identities

Cofunction Identities

Even/Odd Identities

cot��u� � �cot utan��u� � �tan u

sec��u� � sec ucos��u� � cos u

csc��u� � �csc usin��u� � �sin u

csc��

2� u� � sec usec��

2� u� � csc u

cot��

2� u� � tan utan��

2� u� � cot u

cos��

2� u� � sin usin��

2� u� � cos u

1 � cot2 u � csc2 u

1 � tan2 u � sec2 u

sin2 u � cos2 u � 1

cot u �cos u

sin utan u �

sin u

cos u

cot u �1

tan usec u �

1

cos ucsc u �

1

sin u

tan u �1

cot ucos u �

1

sec usin u �

1

csc u

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Section 5.1 Using Fundamental Identities 351

Using the Fundamental IdentitiesOne common use of trigonometric identities is to use given values of trigonometricfunctions to evaluate other trigonometric functions.

Example 1 Using Identities to Evaluate a Function

Use the values and to find the values of all six trigonometric

functions.

SolutionUsing a reciprocal identity, you have

Using a Pythagorean identity, you have

Pythagorean identity

Substitute for cos

Evaluate power.

Simplify.

Because and it follows that lies in Quadrant III. Moreover,because is negative when is in Quadrant III, you can choose the negative rootand obtain Now, knowing the values of the sine and cosine, you canfind the values of all six trigonometric functions.

Now try Exercise 13.

Example 2 Simplifying a Trigonometric Expression

Simplify

SolutionFirst factor out a common monomial factor and then use a fundamental identity.

Factor out monomial factor.

Distributive Property


Multiply.


� �sin3 x

� �sin x�sin2 x�

� �sin x�1 � cos2 x�

sin x cos2 x � sin x � sin x�cos2 x � 1�

sin x cos2 x � sin x.

cot u �1

tan u�

2�5

�2�5

5tan u �

sin u

cos u�

��5�3

�2�3�

�5

2

sec u �1

cos u� �

3

2cos u � �

2

3

csc u �1

sin u� �

3�5

� �3�5

5sin u � �

�5

3

sin u � ��5�3.usin u

utan u > 0,sec u < 0

�59

.

� 1 �4

9

u.�23 � 1 � ��2

3�2

sin2 u � 1 � cos2 u

cos u �1

sec u�

1

�3�2� �

2

3.

tan u > 0sec u � �3

2

Technology TipYou can use a graphingutility to check theresult of Example 2.

To do this, enter

and

Select the line style for andthe path style for Now,graph both equations in thesame viewing window. The twographs appear to coincide, sothe expressions appear to beequivalent. Remember that inorder to be certain that twoexpressions are equivalent, youneed to show their equivalencealgebraically, as in Example 2.

−2

2

�−�

y2.y1

y2 � �sin3 x.

y1 � sin x cos2 x � sin x

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When factoring trigonometric expressions, it is helpful to find a polynomial formthat fits the expression, as shown in Example 3.

Example 3 Factoring Trigonometric Expressions

Factor each expression.

a.

b.

Solutiona. Here the expression is a difference of two squares, which factors as

b. This expression has the polynomial form and it factors as


On occasion, factoring or simplifying can best be done by first rewriting theexpression in terms of just one trigonometric function or in terms of sine or cosinealone. These strategies are illustrated in Examples 4 and 5.

Example 4 Factoring a Trigonometric Expression

Factor

SolutionUse the identity

to rewrite the expression in terms of the cotangent.


Combine like terms.

Factor.


Example 5 Simplifying a Trigonometric Expression

Simplify

SolutionBegin by rewriting in terms of sine and cosine.

Quotient identity

Add fractions.


Reciprocal identity


� csc t

�1

sin t

�sin2 t � cos2 t

sin t

sin t � cot t cos t � sin t � �cos t

sin t�cos t

cot t

sin t � cot t cos t.

� �cot x � 2��cot x � 1�

� cot2 x � cot x � 2

csc2 x � cot x � 3 � �1 � cot2 x� � cot x � 3

csc2 x � 1 � cot2 x

csc2 x � cot x � 3.

4 tan2� � tan � � 3 � �4 tan � � 3��tan � � 1�.

ax2 � bx � c

sec2� � 1 � �sec � � 1��sec � � 1).

4 tan2 � � tan � � 3

sec2 � � 1

Technology TipYou can use the tablefeature of a graphingutility to check the

result of Example 5. To do this,enter

and

Now, create a table that showsthe values of and for different values of The values of and appear to be identical, so the expressionsappear to be equivalent. Toshow that the expressions areequivalent, you need to showtheir equivalence algebraically,as in Example 5.

y2y1

x.y2y1

y2 �1

sin x.

y1 � sin x � �cos xsin x� cos x

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The next two examples involve techniques for rewriting expressions in forms thatare used in calculus.

Example 6 Rewriting a Trigonometric Expression

Rewrite so that it is not in fractional form.

SolutionFrom the Pythagorean identity

you can see that multiplying both the numerator and the denominator by willproduce a monomial denominator.

Multiply.


Write as separate fractions.


Reciprocal and quotient identities


Example 7 Trigonometric Substitution

Use the substitution to write as a trigonometricfunction of

SolutionBegin by letting Then you can obtain

Substitute 2 tan for




Figure 5.1 shows the right triangle illustration of the substitution in Example 7. For you have

opp adj and hyp

Try using these expressions to obtain the result shown in Example 7.

� �4 � x2.� 2,� x,

0 < � < ��2,

sec � > 0 for 0 < � < �2 � 2 sec �.

� �4 sec2 �

� �4�1 � tan2 ��

x.� �4 � x2 � �4 � �2 tan ��2

x � 2 tan �.

�.�4 � x2x � 2 tan �, 0 < � < ��2,

� sec2 x � tan x sec x

�1

cos2 x�

sin x

cos x�

1

cos x

�1

cos2 x�

sin x

cos2 x

�1 � sin x

cos2 x

�1 � sin x

1 � sin2 x

1

1 � sin x�

1

1 � sin x�

1 � sin x

1 � sin x

�1 � sin x�

cos2 x � 1 � sin2 x � �1 � sin x��1 � sin x�

1

1 � sin x

θ

x4 + x2

2

= arctan x 2

Figure 5.1

Multiply numerator anddenominator by �1 � sin x�.

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Using Identities to Evaluate a Function In Exercises7–20, use the given values to evaluate (if possible) all sixtrigonometric functions.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

Matching Trigonometric Expressions In Exercises21–26, match the trigonometric expression with one ofthe following.

(a) (b) (c)

(d) 1 (e) (f)

21. 22.

23. 24.

25. 26.

Matching Trigonometric Expressions In Exercises27–32, match the trigonometric expression with one ofthe following.

(a) (b) (c)

(d) (e) (f)

27. 28.

29. 30.

31. 32.

Simplifying a Trigonometric Expression In Exercises33–44, use the fundamental identities to simplify theexpression. Use the table feature of a graphing utility tocheck your result numerically.

33. 34.

35. 36. sec2 x�1 � sin2 x�sin ��csc � � sin ��cos tan cot x sin x

cos2��2� � xcos x

sec2 x � 1

sin2 x

cot x sec xsec4 x � tan4 x

cos2 x�sec2 x � 1�sin x sec x

sec2 x � tan2 xsec2 xsin x tan x

sin2 xtan xcsc x

sin��2� � xcos��2� � x

sin��x�cos��x�

�1 � cos2 x��csc x�cot2 x � csc2 x

tan x csc xsec x cos x

sin x�tan x

cot x�1sec x

sin � > 0tan � is undefined,

cos � < 0csc � is undefined,

tan � < 0sec � � �3,

sin � < 0tan � � 2,

cos x ��24

5csc��x� � �5,

tan x � �2�5

5sin��x� � �

2

3,

cos x �4

5cos��

2� x� �

3

5,

sin � �8

17sec � � �

17

15,

sin � ��26

26cot � � �5,

sec x � �25

24tan x �

7

24,

cos x � ��3

2tan x �

�3

3,

sin � � ��2

2sec � � �2,

tan � ��3

3csc � � 2,

cos x ��3

2sin x �

1

2,

Vocabulary and Concept Check1. Match each function with an equivalent expression. 2. Match each expression with an equivalent expression.

(a) (i) (a) (i)

(b) (ii) (b) (ii)

(c) (iii) (c) (iii)

In Exercises 3–6, fill in the blank to complete the trigonometric identity.

3. _______

4. _______

5. _______

6. _______

Procedures and Problem Solving

tan��u� �

cos��u� �

csc��

2� u� �

cos��

2� u� �

1 � tan2 ucsc2 u1

csc utan u

1 � cos2 usec2 u1

cot ucos u

1 � cot2 usin2 u1

sec usin u

5.1 Exercises See www.CalcChat.com for worked-out solutions to odd-numbered exercises.For instructions on how to use a graphing utility, see Appendix A.

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37. 38.

39. 40.

41. 42.

43. 44.

Factoring Trigonometric Expressions In Exercises45–54, factor the expression and use the fundamentalidentities to simplify. Use a graphing utility to checkyour result graphically.

45. 46.

47. 48.

49. 50.

51. 52.

53.

54.

Simplifying a Trigonometric Expression In Exercises55–58, perform the multiplication and use the fundamental identities to simplify.

55.

56.

57.

58.

Simplifying a Trigonometric Expression In Exercises59–64, perform the addition or subtraction and use thefundamental identities to simplify.

59. 60.

61. 62.

63. 64.

Rewriting a Trigonometric Expression In Exercises65–70, rewrite the expression so that it is not in fractionalform.

65. 66.

67. 68.

69. 70.

Graphing Trigonometric Functions In Exercises 71–74,use a graphing utility to complete the table and graphthe functions in the same viewing window. Make a conjecture about and

71.

72.

73.

74.

Graphing Trigonometric Functions In Exercises 75–78,use a graphing utility to determine which of the sixtrigonometric functions is equal to the expression.

75. 76.

77.

78.

Trigonometric Substitution In Exercises 79–90, use the trigonometric substitution to write the algebraicexpression as a trigonometric function of where

79.

80.

81.

82.

83.

84.

85.

86.

87.

88.

89.

90.

Solving a Trigonometric Equation In Exercises 91–94,use a graphing utility to solve the equation for where

91. 92.

93. 94. tan � � �sec2 � � 1sec � � �1 � tan2 �

cos � � ��1 � sin2 �sin � � �1 � cos2 �

0 � < 2�.�,

x � �5 cos ��5 � x2,

x � �2 sin ��2 � x2,

3x � 5 sec ��9x2 � 25,

4x � 3 sec ��16x2 � 9,

3x � 2 tan ��9x2 � 4,

2x � 3 tan ��4x2 � 9,

x � 2 cos ��4 � x2,

x � 3 sin ��9 � x2,

x � 10 tan ��x2 � 100,

x � 3 sec ��x2 � 9,

x � 2 cos ��64 � 16x2,

x � 5 sin ��25 � x2,

0 < � < �/2.�,

1

2�1 � sin �

cos ��

cos �

1 � sin ��

sec x �cos x

1 � sin x

sin x�cot x � tan x�cos x cot x � sin x

y2 � tan2 x � tan4 xy1 � sec4 x � sec2 x,

y2 �1 � sin x

cos xy1 �

cos x

1 � sin x,

y2 � sec xy1 � cos x � sin x tan x,

y2 � sin xy1 � cos��

2� x�,

y2. y1

tan2 x

csc x � 1

3

sec x � tan x

tan ysin2 y � cos2 y

sin xtan x

5

tan x � sec x

sin2 y

1 � cos y

tan x1 � sec x

�1 � sec x

tan xtan x �

cos x1 � sin x

cos x

1 � sin x�

1 � sin x

cos xtan x �

sec2 x

tan x

1

sec x � 1�

1

sec x � 1

1

1 � cos x�

1

1 � cos x

�5 � 5 sin x��5 � 5 sin x��csc x � 1��csc x � 1��tan x � sec x��tan x � sec x��sin x � cos x�2

sec3 x � sec2 x � sec x � 1

csc3 x � csc2 x � csc x � 1

sec4 x � tan4 xsin4 x � cos4 x

1 � 2 sin2 x � sin4 xtan4 x � 2 tan2 x � 1

csc2 x � 1

csc x � 1cos2 x � 4cos x � 2

sec2 x tan2 x � sec2 xcot2 x � cot2 x cos2 x

1

cot2 x � 1

cos2 y

1 � sin y

cot��

2� x� cos xsin��

2� x� csc x

tan2 �

sec2 �sec � �

sin �

tan �

sec �

csc �

csc x

cot x

x 0.2 0.4 0.6 0.8 1.0 1.2 1.4

y1

y2

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Rewriting Expressions In Exercises 95–100, rewrite theexpression as a single logarithm and simplify the result.(Hint: Begin by using the properties of logarithms.)

95. 96.

97.

98.

99. 100.

Using Identities In Exercises 101–106, show that theidentity is not true for all values of (There are manycorrect answers.)

101. 102.

103. 104.

105. 106.

Using Identities In Exercises 107–110, use the tablefeature of a graphing utility to demonstrate the identityfor each value of

107. (a) (b)

108. (a) (b)

109. (a) (b)

110. (a) (b)

111. (p. 350) The forces actingon an object weighing units on aninclined plane positioned at an angle of with the horizontal are modeled by

where is the coefficient of friction (see figure). Solve the equation for and simplify the result.

112. Rate of Change The rate of change of the functionis given by the expression

Show that this expression can alsobe written as

113. Rate of Change The rate of change of the functionis given by the expression

Show that this expression can alsobe written as

ConclusionsTrue or False? In Exercises 114 and 115, determinewhether the statement is true or false. Justify youranswer.

114. 115.

Evaluating Trigonometric Functions In Exercises117–120, fill in the blanks. (Note: indicates that approaches from the right, and indicates that approaches from the left.)

117. As and

118. As and

119. As and

120. As and

121. Write each of the other trigonometric functions of interms of sin

122. Write each of the other trigonometric functions of interms of cos

Cumulative Mixed ReviewAdding or Subtracting Rational Expressions In Exercises123–126, perform the addition or subtraction and simplify.

123. 124.

125. 126.

Graphing Trigonometric Functions In Exercises127–130, sketch the graph of the function. (Include twofull periods.)

127. 128.

129.

130. f�x� �32

cos�x � �� 3

f �x� �12

cot�x ��

4�

f�x� � �2 tan �x2

f �x� �12

sin �x

xx2 � 25

�x2

x � 52x

x2 � 4�

7x � 4

6xx � 4

�3

4 � x1

x � 5�

xx � 8

�.�

�.�

csc x →�.x → ��, sin x →�

cot x →�.x →�

2

�

, tan x →�

sec x →�.x → 0�, cos x →�

csc x →�.x →�

2

�

, sin x →�

cxx → c�cxx → c�

cos � sec � � 1sin � csc � � 1

sin x tan2 x.sec x tan x � sin x.f �x� � sec x � cos x

cos x cot2 x.csc x cot x � cos x.f �x� � �csc x � sin x

W

θ

��

�W cos � � W sin �

�W

� �12� � 250 sin�� sin �,

� � 0.8� � 80 cos��

2� �� sin �,

� � 3.1� � 346 tan2 � � 1 � sec2 �,

� �2�

7� � 132 csc2 � � cot2 � � 1,

�.

cot � � �csc2 � � 1csc � � �1 � cot2 �

sec � � �1 � tan2 �sin � � �1 � cos2 �

tan � � �sec2 � � 1cos � � �1 � sin2 �

�.

lncot x � lnsin xlnsec x � lnsin xlncot t � ln�1 � tan2 t� ln�1 � sin x� � lnsec x

lncsc � � lntan �lncos � � lnsin �


116. C A P S T O N E

(a) Use the definitions of sine and cosine to derivethe Pythagorean identity

(b) Use the Pythagorean identity to derive the other Pythagorean identities,

and Discuss how to remember these identities andother fundamental identities.

1 � cot2 � � csc2 �.1 � tan2 � � sec2 �

sin2 � � cos2 � � 1

sin2 � � cos2 � � 1.

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Section 5.2 Verifying Trigonometric Identities 357

5.2 Verifying Trigonometric Identities

What you should learn● Verify trigonometric identities.

Why you should learn itYou can use trigonometric identities

to rewrite trigonometric expressions.

For instance, Exercise 77 on page

364 shows you how trigonometric

identities can be used to simplify an

equation that models the length of a

shadow cast by a gnomon (a device

used to tell time).

Verifying Trigonometric IdentitiesIn this section, you will study techniques for verifying trigonometric identities. In thenext section, you will study techniques for solving trigonometric equations. The key toboth verifying identities and solving equations is your ability to use the fundamentalidentities and the rules of algebra to rewrite trigonometric expressions.

Remember that a conditional equation is an equation that is true for only some ofthe values in its domain. For example, the conditional equation

Conditional equation

is true only for

where is an integer. When you find these values, you are solving the equation. On the other hand, an equation that is true for all real values in the domain of the

variable is an identity. For example, the familiar equation

Identity

is true for all real numbers So, it is an identity.Verifying that a trigonometric equation is an identity is quite different from solving

an equation. There is no well-defined set of rules to follow in verifying trigonometricidentities, and the process is best learned by practice.

Verifying trigonometric identities is a useful process when you need to convert atrigonometric expression into a form that is more useful algebraically. When you verify an identity, you cannot assume that the two sides of the equation are equalbecause you are trying to verify that they are equal. As a result, when verifying identities,you cannot use operations such as adding the same quantity to each side of the equation or cross multiplication.

x.

sin2 x � 1 � cos2 x

n

x � n�

sin x � 0

Guidelines for Verifying Trigonometric Identities

1. Work with one side of the equation at a time. It isoften better to work with the more complicatedside first.

2. Look for opportunities to factor an expression,add fractions, square a binomial, or create a monomial denominator.

3. Look for opportunities to use the fundamental identities. Note which functions are in the finalexpression you want. Sines and cosines pair upwell, as do secants and tangents, and cosecants and cotangents.

4. When the preceding guidelines do not help, tryconverting all terms to sines and cosines.

5. Always try something. Even making an attempt that leads to a dead end provides insight.

Mark Grenier 2010/used under license from Shutterstock.comAndresr 2010/used under license from Shutterstock.com

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Technology TipAlthough a graphingutility can be useful inhelping to verify an

identity, you must use algebraictechniques to produce a validproof. For example, graph the two functions

in a trigonometric viewingwindow. On some graphingutilities the graphs appear to be identical. However,sin 50x � sin 2x.

y2 � sin 2x

y1 � sin 50x


Example 1 Verifying a Trigonometric Identity

Verify the identity.

SolutionBecause the left side is more complicated, start with it.


Simplify.

Reciprocal identity

Quotient identity

Simplify.


There can be more than one way to verify an identity. Here is another way to verify the identity in Example 1.

Rewrite as the difference of fractions.

Reciprocal identity


Remember that an identity is true only for all real values in the domain of the variable. For instance, in Example 1 the identity is not true when because

is not defined when � � ��2.sec2 �� 2

� sin2 �

� 1 � cos2 �

sec2 � � 1

sec2 ��

sec2 �

sec2 ��

1

sec2 �

� sin2 �

�sin2 �cos2 �

�cos2��

� tan2 ��cos2 ��

�tan2 �

sec2 �

sec2 � � 1

sec2 ��

�tan2 � � 1� � 1

sec2 �

sec2 � � 1

sec2 �� sin2 �

Example 2 Combining Fractions Before Using Identities


2 sec2 � �1

1 � sin ��

1

1 � sin �

Algebraic SolutionThe right side is more complicated, so start with it.

Add fractions.

Simplify.


Reciprocal identity


� 2 sec2 �

�2

cos2 �

�2

1 � sin2 �

1

1 � sin ��

1

1 � sin ��

1 � sin � � 1 � sin �

�1 � sin ��1 � sin ��

Numerical SolutionUse a graphing utility to create a table thatshows the values of and

for differentvalues of as shown in Figure 5.2.

Figure 5.2

The values appearto be identical, sothe equationappears to be anidentity.

x,y2 � 1/�1 � sin x� � 1��1 � sin x�

y1 � 2�cos2 x

Michal Bednarek 2010/used under license from Shutterstock.comJason Stitt 2010/used under license from Shutterstock.com

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In Example 2, you needed to write the Pythagorean identity inthe equivalent form

When verifying identities, you may find it useful to write Pythagorean identities in oneof these equivalent forms.

Pythagorean Identities Equivalent Forms

cot2 u � csc2 u � 1

1 � csc2 u � cot2 u

tan2 u � sec2 u � 1

1 � sec2 u � tan2 u

cos2 u � 1 � sin2 u

sin2 u � 1 � cos2 u

cos2 u � 1 � sin2 u.


1 � cot2 u � csc2 u

1 � tan2 u � sec2 u



Verify the identity �tan2 x � 1��cos2 x � 1� � �tan2 x.

Algebraic SolutionBy applying identities before multiplying, you obtain the following.

Pythagorean identities

Reciprocal identity

Property of exponents

Quotient identity


� �tan2 x

� �� sin x

cos x�2

� �sin2 x

cos2 x

�tan2 x � 1��cos2 x � 1� � �sec2 x��sin2 x�

Graphical Solution

−3

2

� −2 2�

y2 = − tan2 x

y1 = (tan2 x + 1)(cos2 x − 1)

Because the graphs appear tocoincide, the given equationappears to be an identity.

Example 4 Converting to Sines and Cosines

Verify the identity

SolutionIn this case there appear to be no fractions to add, no products to find, and no opportunities to use the Pythagorean identities. So, try converting the left side to sinesand cosines.

Quotient identities

Add fractions.


Product of fractions

Reciprocal identities


� sec x csc x

�1

cos x

1

sin x

�1

cos x sin x

�sin2 x � cos2 x

cos x sin x

tan x � cot x �sin x

cos x�

cos x

sin x

tan x � cot x � sec x csc x.

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Recall from algebra that rationalizing the denominator using conjugates is, onoccasion, a powerful simplification technique. A related form of this technique worksfor simplifying trigonometric expressions as well. For instance, to simplify

multiply the numerator and the denominator by

As shown above, is considered a simplified form of

because the expression does not contain any fractions.

11 � cos x

csc2 x�1 � cos x�

� csc2 x�1 � cos x�

�1 � cos x

sin2 x

�1 � cos x1 � cos2 x

1

1 � cos x�

11 � cos x�

1 � cos x1 � cos x�

�1 � cos x�.

11 � cos x

In Examples 1 through 5, you have been verifying trigonometric identities byworking with one side of the equation and converting it to the form given on the otherside. On occasion it is practical to work with each side separately to obtain one common form equivalent to both sides. This is illustrated in Example 6.



sec x � tan x �cos x

1 � sin x

Algebraic SolutionBegin with the right side because you can create a monomial denominatorby multiplying the numerator and denominator by

Multiply.



Simplify.

Identities


� sec x � tan x

�1

cos x�

sin x

cos x

�cos x

cos2 x�

cos x sin x

cos2 x

�cos x � cos x sin x

cos2 x

�cos x � cos x sin x

1 � sin2 x

cos x

1 � sin x�

cos x

1 � sin x �1 � sin x

1 � sin x��1 � sin x�.

Graphical Solution

−5

5

7�2

9�2

−

y1 = sec x + tan x

cos x1 − sin x

y2 =

Because the graphs appear tocoincide, the given equationappears to be an identity.

Multiply numerator and denominator by �1 � sin x�.

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In Example 7, powers of trigonometric functions are rewritten as more complicatedsums of products of trigonometric functions. This is a common procedure used in calculus.

Example 7 Examples from Calculus

Verify each identity.

a.

b.

c.

Solutiona. Write as separate factors.


Multiply.

b. Write as separate factors.


Multiply.

c. Write as separate factors.


Multiply.


� csc2 x�cot x � cot3 x�

� csc2 x�1 � cot2 x� cot x

csc4 x cot x � csc2 x csc2 x cot x

� �cos4 x � cos6 x�sin x

� �1 � cos2 x�cos4 x sin x

sin3 x cos4 x � sin2 x cos4 x sin x

� tan2 x sec2 x � tan2 x

� tan2 x�sec2 x � 1�

tan4 x � �tan2 x��tan2 x�

csc4 x cot x � csc2 x�cot x � cot3 x�sin3 x cos4 x � �cos4 x � cos6 x�sin x

tan4 x � tan2 x sec2 x � tan2 x

Example 6 Working with Each Side Separately


cot2 �

1 � csc ��

1 � sin �

sin �

Algebraic SolutionWorking with the left side, you have


Factor.

Simplify.

Now, simplifying the right side, you have


Reciprocal identity

The identity is verified because both sides are equal to


csc � � 1.

� csc � � 1.

1 � sin �

sin ��

1

sin ��

sin �

sin �

� csc � � 1.

��csc � � 1��csc � � 1�

1 � csc �

cot2 �

1 � csc ��

csc2 � � 1

1 � csc �

Numerical SolutionUse a graphing utility to create a table that shows thevalues of

and

for different values of as shown in Figure 5.3.

Figure 5.3

The values for y1and y2 appear tobe identical, so theequation appears tobe an identity.

x,

y2 �1 � sin x

sin x

y1 �cot2 x

1 � csc x

What’s Wrong?To determine the validity of the statement

you use

a graphing utility to graphand

as shown in the

figure. You use the graph to conclude that the statement is anidentity. What’s wrong?

−20

−3� 3�

20

y2 �56

tan2 x,

y1 � tan2 x sin2 x

tan2 x sin2 x �? 5

6 tan2 x,

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Verifying a Trigonometric Identity In Exercises 11–20,verify the identity.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

Algebraic-Graphical-Numerical In Exercises 21–28, usea graphing utility to complete the table and graph thefunctions in the same viewing window. Use both the tableand the graph as evidence that Then verify theidentity algebraically.

21.

22.

23.

24.

25.

26.

27.

28.

Error Analysis In Exercises 29 and 30, describe theerror.

29.

30.

�1

sin �� csc �

�1 � sec �

sin ��1 � sec ��

�1 � sec �

�sin ��1 � � 1cos ��

1 � sec��

sin�� tan�� 1 � sec �

sin � � tan �

� 2 � cot x � tan x

� 1 � cot x � tan x � 1

� 1 � cot x � tan x � tan x cot x

� �1 � tan x��1 � cot x� �1 � tan x�1 � cot��x�

y2 � csc x � sin xy1 �1

sin x�

1

csc x,

y2 � tan x � cot xy1 �1

tan x�

1

cot x,

y2 � sec xy1 � cos x � sin x tan x,

y2 � csc xy1 � sin x � cos x cot x,

y2 � sin x tan xy1 � sec x � cos x,

y2 � cos x cot xy1 � csc x � sin x,

y2 � csc xy1 �csc x � 1

1 � sin x,

y2 � csc x � sin xy1 �1

sec x tan x,

y1 � y2.

tan2 y�csc2 y � 1� � 1

�1 � sin x��1 � sin x� � cos2 x

2 � csc2 z � 1 � cot2 z

tan2 � � 6 � sec2 � � 5

cos2 � sin2 � 2 cos2 � 1

cos2 � sin2 � 1 � 2 sin2

sin2 ttan2 t

� cos2 t

csc2 x

cot x� csc x sec x

sec y cos y � 1

sin t csc t � 1

Vocabulary and Concept CheckIn Exercises 1–8, fill in the blank to complete the trigonometric identity.

1. _______ 2. _______

3. _______ 4. _______

5. _______ 6. _______

7. _______ 8. _______

9. Is a graphical solution sufficient to verify a trigonometric identity?

10. Is a conditional equation true for all real values in its domain?


sec��u� �sin��u� �

tan��

2� u� �� 1sin2 u �

1sec u

�sin ucos u

�

1csc u

�1

tan u�


x 0.2 0.4 0.6 0.8 1.0 1.2 1.4

y1

y2

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Verifying a Trigonometric Identity In Exercises 31 and32, fill in the missing step(s).

31.

32.


33.

34.

35.

36.

37.

38.

Verifying a Trigonometric Identity In Exercises 39–46,verify the identity algebraically. Use the table feature ofa graphing utility to check your result numerically.

39.

40.

41.

42.

43.

44.

45.

46.

Verifying a Trigonometric Identity In Exercises 47–58,verify the identity algebraically. Use a graphing utility tocheck your result graphically.

47.

48.

49.

50.

51.

52.

53.

54.

55.

56.

57.

58.

Graphing a Trigonometric Function In Exercises 59–62,use a graphing utility to graph the trigonometric function. Use the graph to make a conjecture about asimplification of the expression. Verify the resultingidentity algebraically.

59.

60.

61. 62.

Verifying an Identity Involving Logarithms In Exercises63 and 64, use the properties of logarithms and trigonometric identities to verify the identity.

63.

64.

Using Cofunction Identities In Exercises 65–68, use thecofunction identities to evaluate the expression withoutusing a calculator.

65. 66.

67.

68. sin2 18� � sin2 40� � sin2 50� � sin2 72�

cos2 20� � cos2 52� � cos2 38� � cos2 70�

cos2 14� � cos2 76�sin2 35� � sin2 55�

ln�sec �� ln�cos ��ln�cot �� ln�cos �� ln�sin ��

y � sin t �cot2 t

csc ty �

1

sin x�

cos2 x

sin x

y �cos x

1 � tan x�

sin x cos x

sin x � cos x

y �1

cot x � 1�

1

tan x � 1

sin3 � cos3

sin � cos � 1 � sin cos

tan3 � � 1

tan � � 1� tan2 � � tan � � 1

cot �

csc � � 1�

csc � � 1

cot �

sin

1 � cos �

1 � cos

sin

tan �1 � sec �

�1 � sec �

tan �� 2 csc �

1 �sin2 �

1 � cos �� cos �

sin � csc � � sin2 � � cos2 �

csc � tan � � sec �

1 � csc �sec �

� cot � � cos �

cot x tan xsin x

� csc x

csc x�csc x � sin x��sin x � cos x

sin x� cot x � csc2 x

2 sec2 x � 2 sec2 x sin2 x � sin2 x � cos2 x � 1

sec2��

2� x� � 1 � cot2 x

sin x csc��

2� x� � tan x

sec2 y � cot2��

2� y� � 1

sin2��

2� x� � sin2 x � 1

�sec � � tan ��csc � � 1� � cot �

cos �1 � sin �

� sec � � tan �

tan x � cot y

tan x cot y� tan y � cot x

cos x � cos y

sin x � sin y�

sin x � sin y

cos x � cos y� 0

�1 � sin y�1 � sin��y� � cos2 y

csc��x�sec��x�

� �cot x

sec��2� � xtan��2� � x

� sec x

cot��

2� x�csc x � sec x

sec6 x�sec x tan x� �sec4 x�sec x tan x� � sec5 x tan3 x

sin1�2 x cos x � sin5�2 x cos x � cos3 x�sin x

� 1 � 2 cos2 x

�� sin2 x � cos2 x

�sin2 x � cos2 x

1

��

tan x � cot xtan x � cot x

�

sin xcos x

�cos xsin x

sin xcos x

�cos xsin x

� tan4 x

�� sec4 x � 2 sec2 x � 1 � �sec2 x � 1�2

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Examples from Calculus In Exercises 69–72, powers oftrigonometric functions are rewritten to be useful in calculus. Verify the identity.

69.

70.

71.

72.


73.

74.

75.

76.

77. (p. 357) The length of ashadow cast by a vertical gnomon (a deviceused to tell time) of height when the angleof the sun above the horizon is (see figure)can be modeled by the equation

(a) Verify that the equation for is equal to

(b) Use a graphing utility to complete the table. Letfeet.

(c) Use your table from part (b) to determine the anglesof the sun that result in the maximum and minimum lengths of the shadow.

(d) Based on your results from part (c), what time ofday do you think it is when the angle of the sunabove the horizon is

78. Rate of Change The rate of change of the functionis given by

Show that the expression for the rate of change can alsobe given by

ConclusionsTrue or False? In Exercises 79–82, determine whetherthe statement is true or false. Justify your answer.

79. There can be more than one way to verify a trigonometricidentity.

80. Of the six trigonometric functions, two are even.

81. The equation is an identity,because and

82.

Verifying a Trigonometric Identity In Exercises 83and 84, (a) verify the identity and (b) determine whetherthe identity is true for the given value of Explain.

83.

84.

Using Trigonometric Substitution In Exercises 85–88,use the trigonometric substitution to write the algebraicexpression as a trigonometric function of where

Assume

85.

86.

87.

88.

Think About It In Exercises 89–92, explain why theequation is not an identity and find one value of the variable for which the equation is not true.

89. 90.

91.

92.

93. Verify that for all integers

Cumulative Mixed ReviewGraphing an Exponential Function In Exercises 95–98,use a graphing utility to construct a table of values forthe function. Then sketch the graph of the function.Identify any asymptotes of the graph.

95. 96.

97. 98. f�x� � 2x�1 � 3 f�x� � 2�x � 1

f�x� � �2x�3 f�x� � 2x � 3

cos��2n � 1��2 � � 0.n,

�sin2 x � cos2 x � sin x � cos x

tan � � �sec2 � � 1

sin � � �1 � cos2 ��tan2 x � tan x

u � a sec ��u2 � a2,

u � a tan ��a2 � u2,

u � a cos ��a2 � u2,

u � a sin ��a2 � u2,

a > 0.0 < � < �/2.�,

x � �sec xtan x

�tan x

sec x � cos x,

x � 0sin x

1 � cos x�

1 � cos xsin x

,

x.

sin�x2� � sin2�x�1 � tan2�0� � 1.sin2�0� � cos2�0� � 1

sin2 � � cos2 � � 1 � tan2 �

�cos x cot2 x.

cos x � csc x cot x.f�x� � sin x � csc x

90�?

h � 5

h cot �.s

h ft

sθ

s �h sin�90� � ��

sin �.

�h

s

tan�cos�1 x � 1

2 � ��4 � �x � 1�2

x � 1

tan�sin�1 x � 1

4 � �x � 1

�16 � �x � 1�2

cos�sin�1 x� � �1 � x2

tan�sin�1 x� �x

�1 � x2

sin4 x � cos4 x � 1 � 2 cos2 x � 2 cos4 x

cos3 x sin2 x � �sin2 x � sin4 x�cos x

sec4 x tan2 x � �tan2 x � tan4 x�sec2 x

tan5 x � tan3 x sec2 x � tan3 x

� 15� 30� 45� 60� 75� 90�

s94. C A P S T O N E Write a study sheet explaining the

difference between a trigonometric identity and a conditional equation. Include suggestions on how toverify a trigonometric identity.

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Section 5.3 Solving Trigonometric Equations 365

IntroductionTo solve a trigonometric equation, use standard algebraic techniques such as collectinglike terms and factoring. Your preliminary goal is to isolate the trigonometric functioninvolved in the equation.

Example 1 Solving a Trigonometric Equation

Solve

Solution

Write original equation.

Add 1 to each side.

Divide each side by 2.

To solve for note in Figure 5.4 that the equation has solutions andin the interval Moreover, because has a period of there are

infinitely many other solutions, which can be written as

and General solution

where is an integer, as shown in Figure 5.4.

Figure 5.4


Another way to show that the equation has infinitely many solutions isindicated in Figure 5.5. Any angles that are coterminal with or are also solutions of the equation.

Figure 5.5

( ( π 5 6 π

6

π π 5 1 2 6

sin + 2n = ( ( π π 1 2 6

sin + 2n =

5��6��6sin x �

12

x

π π

π 1

−1

π π −

5 6

π 6

x = − 2

x = − 2

π π 5 6

x = + 2 π 5 6

x =

π π 6

x = + 2 π 6

x =

y = sin x

y

y = 12

n

x �5�

6� 2n�x �

�

6� 2n�

2�,sin x�0, 2��.x � 5��6x � ��6sin x �

12x,

sin x �12

2 sin x � 1

2 sin x � 1 � 0

2 sin x � 1 � 0.

5.3 Solving Trigonometric Equations

What you should learn● Use standard algebraic

techniques to solve trigonometric

equations.

● Solve trigonometric equations

of quadratic type.

● Solve trigonometric equations

involving multiple angles.

● Use inverse trigonometric

functions to solve trigonometric

equations.

Why you should learn itYou can use trigonometric equations

to solve a variety of real-life problems.

For instance, in Exercise 96 on page

375, you can solve a trigonometric

equation to help answer questions

about monthly sales of skis.

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Example 3 Extracting Square Roots

Solve

SolutionRewrite the equation so that is isolated on one side of the equation.


Add 1 to each side.


Extract square roots.

Rationalize the denominator.

Because has a period of first find all solutions in the interval These are

and

Finally, add multiples of to each of these solutions to get the general form


where is an integer. You can confirm this answer by graphing witha graphing utility, as shown in Figure 5.7. The graph has -intercepts at

and so on. These -intercepts correspond to the solutions of


3 tan2 x � 1 � 0.x7��6,5��6,��6,x

y � 3 tan2 x � 1n

x �5�

6� n�x �

�

6� n�

�

x �5�

6.x �

�

6

�0, ��.�,tan x

tan x � ±�33

tan x � ±1�3

tan2 x �1

3

3 tan2 x � 1

3 tan2 x � 1 � 0

tan x

3 tan2 x � 1 � 0.

Example 2 Collecting Like Terms

Find all solutions of

in the interval �0, 2��.

sin x � �2 � �sin x

Algebraic SolutionRewrite the equation so that is isolated on one side ofthe equation.


Combine like terms.


The solutions in the interval are

and


x �7�

4.x �

5�

4

�0, 2��

sin x � ��2

2

2 sin x � ��2

sin x � sin x � ��2

sin x � �2 � �sin x

sin x

Numerical SolutionUse a graphing utility set in radian mode to create a table thatshows the values of and for different values of Your table should go from to

using increments of as shown in Figure 5.6.

Figure 5.6

The values of y1 and y2appear to be identicalwhen x ≈ 3.927 ≈ 5 /4and x ≈ 5.4978 ≈ 7 /4.These values are theapproximate solutionsof sin x + 2 = −sin x.

ππ

��8,x � 2�

x � 0x.y2 � �sin xy1 � sin x � �2

−2

6 y = 3 tan2 x − 1

�2

5�2

−

Figure 5.7

Add sin to and subtractfrom each side.�2

x

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The equations in Examples 1–3 involved only one trigonometric function. Whentwo or more functions occur in the same equation, collect all terms on one side and tryto separate the functions by factoring or by using appropriate identities. This may produce factors that yield no solutions, as illustrated in Example 4.

Example 4 Factoring

Solve

SolutionBegin by rewriting the equation so that all terms are collected on one side of the equation.


Subtract 2 cot from each side.

Factor.

By setting each of these factors equal to zero, you obtain the following.

and

In the interval the equation has the solution

No solution is obtained for

because are outside the range of the cosine function. Because has a periodof the general form of the solution is obtained by adding multiples of to to get

General solution

where is an integer. The graph of (in dot mode), shown inFigure 5.8, confirms this result. From the graph you can see that the intercepts occur at

and so on. These intercepts correspond to the solutions of

Figure 5.8


−3

3

�−� 3

y = cot x cos2 x − 2 cot x

cot x cos2 x � 2 cot x.

x-

��

2, �

2, 3�

2, 5�

2

x-y � cot x cos2 x � 2 cot xn

x ��

2� n�

x � ��2,��,cot x±�2

cos x � ±�2

x ��

2.

cot x � 0�0, ��,

cos x � ±�2

cos2 x � 2

cos2 x � 2 � 0 cot x � 0

cot x�cos2 x � 2� � 0

x cot x cos2 x � 2 cot x � 0

cot x cos2 x � 2 cot x

cot x cos2 x � 2 cot x.

Explore the ConceptUsing the equation inExample 4, explainwhat happens when

each side of the equation isdivided by Why is this an incorrect method to usewhen solving an equation?

cot x.

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Equations of Quadratic TypeMany trigonometric equations are of quadratic type as shownbelow. To solve equations of this type, factor the quadratic or, when factoring is not possible, use the Quadratic Formula.

Quadratic in sin x Quadratic in sec x

�sec x�2 � 3 �sec x� � 2 � 0 2�sin x�2 � sin x � 1 � 0

sec2 x � 3 sec x � 2 � 0 2 sin2 x � sin x � 1 � 0

ax2 � bx � c � 0,

Example 5 Factoring an Equation of Quadratic Type

Find all solutions of in the interval �0, 2��.2 sin2 x � sin x � 1 � 0

Algebraic SolutionTreating the equation as a quadratic in and factoringproduces the following.


Factor.

Setting each factor equal to zero, you obtain the followingsolutions in the interval

and


x ��

2x �

7�

6,

11�

6

sin x � 1sin x � �1

2

sin x � 1 � 02 sin x � 1 � 0

�0, 2��.

�2 sin x � 1��sin x � 1� � 0

2 sin2 x � sin x � 1 � 0

sin x

Graphical Solution

Figure 5.9

From Figure 5.9, you can conclude that the approximate solutions of in the interval are

and x � 5.760 �11�

6.x � 3.665 �

7�

6,x � 1.571 �

�

2,

�0, 2��2 sin2 x � sin x � 1 � 0

−2

3

�20

y = 2 sin2 x − sin x − 1

The x-intercepts arex ≈ 1.571, x ≈ 3.665,and x ≈ 5.760.

When working with an equation of quadratic type, be sure that the equationinvolves a single trigonometric function, as shown in the next example.

Example 6 Rewriting with a Single Trigonometric Function

Solve

SolutionBegin by rewriting the equation so that it has only cosine functions.



Factor.

By setting each factor equal to zero, you can find the solutions in the interval tobe and Because cos has a period of the general solution is

General solution

where is an integer. The graph of shown in Figure 5.10,confirms this result.


y � 2 sin2 x � 3 cos x � 3,n

x �5�

3� 2n�x �

�

3� 2n�,x � 2n�,

2�,xx � 5��3.x � ��3,x � 0,�0, 2��

�2 cos x � 1��cos x � 1� � 0

2 cos2 x � 3 cos x � 1 � 0

2�1 � cos2 x� � 3 cos x � 3 � 0

2 sin2 x � 3 cos x � 3 � 0

2 sin2 x � 3 cos x � 3 � 0.

Combine like terms andmultiply each side by �1.

−6

1

�2�3

−

y = 2 sin2 x + 3 cos x − 3

Figure 5.10

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Sometimes you must square each side of an equation to obtain a quadratic. Becausethis procedure can introduce extraneous solutions, you should check any solutions inthe original equation to see whether they are valid or extraneous.

Example 7 Squaring and Converting to Quadratic Type

Find all solutions of in the interval

SolutionIt is not clear how to rewrite this equation in terms of a single trigonometric function.Notice what happens when you square each side of the equation.


Square each side.


Combine like terms.

Factor.

Setting each factor equal to zero produces the following.

and

Because you squared the original equation, check for extraneous solutions.

Check

Substitute for

Solution checks. ✓

Substitute for

Solution does not check.

Substitute for

Solution checks. ✓

Of the three possible solutions, is extraneous. So, in the interval theonly solutions are and The graph of shown in Figure 5.11, confirms this result because the graph has two -intercepts

in the interval

Figure 5.11


−1

3

�20

y = cos x + 1 − sin x

�0, 2��.�at x � ��2 and x � ��x

y � cos x � 1 � sin x,x � �.x � ��2�0, 2��,x � 3��2

�1 � 1 � 0

x.� cos � � 1 �?

sin �

0 � 1 � �1

x.3��2 cos 3�

2� 1 �

?sin

3�

2

0 � 1 � 1

x.��2 cos �

2� 1 �

?sin

�

2

x � � x ��

2,

3�

2

cos x � �1 cos x � 0

cos x � 1 � 0 2 cos x � 0

2 cos x�cos x � 1� � 0

2 cos2 x � 2 cos x � 0

cos2 x � 2 cos x � 1 � 1 � cos2 x

cos2 x � 2 cos x � 1 � sin2 x

cos x � 1 � sin x

�0, 2��.cos x � 1 � sin x Explore the ConceptUse a graphing utilityto confirm the solutionsfound in Example 7

in two different ways. Do bothmethods produce the same -values? Which method do

you prefer? Why?

1. Graph both sides of theequation and find the -coordinates of the points

at which the graphs intersect.

Left side:

Right side:

2. Graph the equationand

find the -intercepts of thegraph.

xy � cos x � 1 � sin x

y � sin x

y � cos x � 1

x

x

Edyta Pawlowska 2010/used under license from Shutterstock.com

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Functions Involving Multiple AnglesThe next two examples involve trigonometric functions of multiple angles of the forms

and To solve equations of these forms, first solve the equation for thendivide your result by

Example 8 Functions of Multiple Angles

Solve

Solution


Add 1 to each side.


In the interval you know that and are the only solutions.So in general, you have and Dividing this resultby 3, you obtain the general solution


where is an integer. This solution is confirmed graphically in Figure 5.12.


Example 9 Functions of Multiple Angles

Original equation

Subtract 3 from each side.


In the interval you know that is the only solution. So in general, youhave Multiplying this result by 2, you obtain the general solution

General solution


Figure 5.13


−20

20

�−2 2�

y = 3 tan + 3x2

n

x �3�

2� 2n�

x�2 � 3��4 � n�.x�2 � 3��4�0, ��,

tan x

2� �1

3 tan x

2� �3

3 tan x

2� 3 � 0

n

t �5�

9�

2n�

3t �

�

9�

2n�

3

3t � 5��3 � 2n�.3t � ��3 � 2n�3t � 5��33t � ��3�0, 2��,

cos 3t �1

2

2 cos 3t � 1

2 cos 3t � 1 � 0

2 cos 3t � 1 � 0.

k.ku,cos ku.sin ku

−4

2

�

y = 2 cos 3t − 1

20

Figure 5.12

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Using Inverse Functions

Example 10 Using Inverse Functions

Solve

Solution



Combine like terms.

Factor.

Setting each factor equal to zero, you obtain two solutions in the interval [Recall that the range of the inverse tangent function is

and

Finally, because tan has a period of add multiples of to obtain




With some trigonometric equations, there is no reasonable way to find the solutions algebraically. In such cases, you can still use a graphing utility to approximatethe solutions.

Example 11 Approximating Solutions

Approximate the solutions of

in the interval

SolutionUse a graphing utility to graph in the interval Using the zeroor root feature, you can see that the solutions are

and See Figure 5.15.

Figure 5.15


y � x � 2 sin x

−3

3

�−�

−3

3

�−�

x � 1.8955.x � 0,x � �1.8955,

��, ��.y � x � 2 sin x

��, ��.

x � 2 sin x

n

x � ��

4� n�x � arctan 3 � n�

��,x

x � arctan��1� � ��

4 x � arctan 3

tan x � �1 tan x � 3

��2, ��2�.]��2, ��2�.

�tan x � 3��tan x � 1� � 0

tan2 x � 2 tan x � 3 � 0

1 � tan2 x � 2 tan x � 4 � 0

sec2 x � 2 tan x � 4

sec2 x � 2 tan x � 4.

−3

3

�−�

−4

6

�2

�2

−

y = sec2 x − 2 tan x − 4

Figure 5.14

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Example 12 Surface Area of a Honeycomb

The surface area of a honeycomb is given by the equation

where inches, inch, and is the angle indicated in Figure 5.16.

a. What value of gives a surface area of 12 square inches?

b. What value of gives the minimum surface area?

Solutiona. Let and

Using a graphing utility set in degree mode, you can graph the function

Using the zero or root feature, you can determine that

and See Figure 5.17.

Figure 5.17

b. From part (a), let to obtain

Graph this function using a graphing utility set in degree mode. Use the minimumfeature to approximate the minimum point on the graph, which occurs at

as shown in Figure 5.18. By using calculus, it can be shown that the exact minimumvalue is


� � arccos 1�3 � 54.7356.

� � 54.7

S � 10.8 � 0.84375�3 � cos �sin � .

h � 2.4 and s � 0.75

y � 0.84375��3 � cos xsin x � � 1.2

−0.02

0.05

0 90

−0.02

0.05

0 90

� � 59.9.� � 49.9

y � 0.84375�3 � cos xsin x � 1.2.

0 � 0.84375�3 � cos �

sin � � 1.2

12 � 10.8 � 0.84375�3 � cos �

sin �

12 � 6�2.4��0.75� �32

�0.75�2�3 � cos �sin �

S � 6hs �32

s2�3 � cos �sin �

S � 12.h � 2.4, s � 0.75,

�

�

�s � 0.75h � 2.4

0 < � � 90S � 6hs �3

2s2�3 � cos �

sin � ,

11

14

0 150

y = 10.8 + 0.843753 − cos x

sin x ( (

Figure 5.18

s = 0.75 in .

θ

h = 2.4 in .

Figure 5.16

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Verifying a Solution In Exercises 5–10, verify that each -value is a solution of the equation.

5.

(a) (b)

6.

(a) (b)

7.

(a) (b)

8.

(a) (b)

9.

(a) (b)

10.

(a) (b)

Solving a Trigonometric Equation In Exercises 11–16,find all solutions of the equation in the interval

11. 12.

13. 14.

15. 16.

Solving a Trigonometric Equation In Exercises 17–28,find all solutions of the equation in the interval

17. 18.

19. 20.

21. 22.

23. 24.

25. 26.

27. 28.

Solving a Trigonometric Equation In Exercises 29–36,solve the equation.

29. 30.

31. 32.

33. 34.

35. 36.

Solving a Trigonometric Equation In Exercises 37– 48,find all solutions of the equation in the interval algebraically. Use the table feature of a graphing utilityto check your answers numerically.

37. 38.

39. 40.

41. 42.

43. 44.

45. 46.

47.

48.

Approximating Solutions In Exercises 49–56, use agraphing utility to approximate the solutions of theequation in the interval by collecting all terms onone side, graphing the new equation, and using the zero orroot feature to approximate the -intercepts of the graph.

49.

50.

51. 52.

53. 54. 4 sin x � cos x � 2csc x � cot x � 1

csc2 x � 3 csc x � 44 sin2 x � 2 cos x � 1

2 sec2 x � tan2 x � 3 � 0

2 sin2 x � 3 sin x � 1 � 0

x

[0, 2�

sin2 x � cos x � 1 � 0

2 sec2 x � tan2 x � 3 � 0

sec x � tan x � 12 sin x � csc x � 0

sec x csc x � 2 csc xsec2 x � sec x � 2

2 sin2 x � 2 � cos xcos3 x � cos x

tan2 x � 1 � 0csc2 x � 2 � 0

sin x � 1 � 0tan x � �3 � 0

[0, 2�

cos x�cos x � 1� � 04 cos2 x � 1 � 0

3 cot2 x � 1 � 03 sec2 x � 4 � 0

cot x � 1 � 0�3 sec x � 2 � 0

�2 sin x � 1 � 02 sin x � 1 � 0

csc x � ��2tan x � �1

sec x � 2cot x � �3

sec x � �2csc x � �2

cos x ��22

tan x � ��33

sin x ��32

cot x � �1

sin x � �12

cos x � ��32

[0, 2� .

tan x � ��3tan x � 1

sin x � ��22

cos x � �12

cos x ��32

sin x �12

[0�, 360� .

x �5�

6x �

�

6

csc4 x � 4 csc2 x � 0

x �7�

6x �

�

2

2 sin2 x � sin x � 1 � 0

x �7�

8x �

�

8

4 cos2 2x � 2 � 0

x �5�

12x �

�

12

3 tan2 2x � 1 � 0

x �5�

3x �

�

3

sec x � 2 � 0

x �5�

3x �

�

3

2 cos x � 1 � 0

x

Vocabulary and Concept CheckIn Exercises 1 and 2, fill in the blank.

1. The _______ solution of the equation is given by

and where is an integer.

2. The equation is an equation of _______ type.

3. Is a solution of the equation

4. To solve do you divide each side by


sec x?sec x sin2 x � sec x,

cos x � 0?x � 0

tan2 x � 5 tan x � 6 � 0

nx �5�

3� 2n�,x �

�

3� 2n�

2 cos x � 1 � 0


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55.

56.

Graphing a Trigonometric Function In Exercises 57–60,(a) use a graphing utility to graph each function in theinterval (b) write an equation whose solutionsare the points of intersection of the graphs, and (c) usethe intersect feature of the graphing utility to find thepoints of intersection (to four decimal places).

57.

58.

59.

60.

Functions of Multiple Angles In Exercises 61–68, solvethe multiple-angle equation.

61. 62.

63. 64.

65. 66.

67. 68.

Approximating x-Intercepts In Exercises 69–72,approximate the -intercepts of the graph. Use a graphingutility to check your solutions.

69. 70.

71. 72.

Approximating Solutions In Exercises 73–80, use agraphing utility to approximate the solutions of theequation in the interval

73. 74.

75. 76.

77. 78.

79.

80.

Using Inverse Functions In Exercises 81 and 82, useinverse functions where necessary to find all solutions ofthe equation in the interval

81.

82.

Approximating Solutions In Exercises 83–86, use agraphing utility to approximate the solutions (to threedecimal places) of the equation in the given interval.

83.

84.

85.

86.

Approximating Maximum and Minimum Points InExercises 87–92, (a) use a graphing utility to graph thefunction and approximate the maximum and minimumpoints (to four decimal places) of the graph in the interval and (b) solve the trigonometric equationand verify that the -coordinates of the maximum and minimum points of are among its solutions (the trigonometric equation is found using calculus).

Function Trigonometric Equation

87.

88.

89.

90.

91.

92.

Finding a Fixed Point In Exercises 93 and 94, find thesmallest positive fixed point of the function A fixedpoint of a function is a real number such that

93. 94.

95. Economics The monthly unit sales (in thousands)of lawn mowers are approximated by

where is the time (in months), with correspondingto January. Determine the months in which unit salesexceed 100,000.

t � 1t

U � 74.50 � 43.75 cos � t

6

U

f�x� � cos x f�x� � tan �x

4

f �c � c.]cf[f.

2 cos x � 4 sin x cos x � 0 f �x� � 2 sin x � cos 2x

cos x � sin x � 0 f �x� � sin x � cos x

�2 sin x cos x � cos x � 0 f�x� � cos2 x � sin x

2 sin x cos x � sin x � 0 f�x� � sin2 x � cos x

�2 sin 2x � 0 f�x� � cos 2x

2 cos 2x � 0 f�x� � sin 2x

fx

[0, 2��,

��

2,

�

2�2 sec2 x � tan x � 6 � 0,

��

2,

�

2�4 cos2 x � 2 sin x � 1 � 0,

�0, ��cos2 x � 2 cos x � 1 � 0,

��

2,

�

2�3 tan2 x � 5 tan x � 4 � 0,

sec2 x � tan x � 3

tan2 x � 6 tan x � 5 � 0

[0, 2� .

3 tan2 x � 4 tan x � 4 � 0

12 sin2 x � 13 sin x � 3 � 0

csc2 x � 0.5 cot x � 5sec2 x � 0.5 tan x � 1

2x sin x � 2 � 0x tan x � 1 � 0

2 sin x � cos x � 02 cos x � sin x � 0

[0, 2� .

−4

4

−3 3

−4

2

−3 3

y � sec4� x

8 � 4y � tan2�x

6 � 3

−2

2

−1 3

−1

3

−2 4

y � sin � x � cos � xy � sin �x

2� 1

x

tan x

3� 1cos

x

2�

�2

2

sec 4x � 2sin 2x � ��3

2

cos 2x � �1sin 4x � 1

sin x2

� 0cos x4

� 0

y � e�x � x � 1y � cos2 x,

y � ex � 4xy � sin2 x,

y � x � x2y � cos x,

y � x2 � 2xy � sin 2x,

[0, 2� ,

1 � sin xcos x

�cos x

1 � sin x� 4

cos x cot x1 � sin x

� 3

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96. (p. 365) The monthly unitsales (in hundreds) of skis for a chain ofsports stores are approximated by

where is the time (in months), with corresponding to January. Determine themonths in which unit sales exceed 7500.

97. Exploration Consider the function

and its graph shown in the figure.

(a) What is the domain of the function?

(b) Identify any symmetry or asymptotes of the graph.

(c) Describe the behavior of the function as

(d) How many solutions does the equationhave in the interval Find the

solutions.

(e) Does the equation have a greatestsolution? If so, approximate the solution. If not,explain.


and its graph shown in the figure.

(a) What is the domain of the function?

(b) Identify any symmetry or asymptotes of the graph.

(c) Describe the behavior of the function as

(d) How many solutions does the equationhave in the interval Find the

solutions.

99. Harmonic Motion A weight is oscillating on the endof a spring (see figure). The position of the weight relative to the point of equilibrium is given by

where is the displacement (in meters) and is thetime (in seconds). Find the times at which the weightis at the point of equilibrium for

100. Damped Harmonic Motion The displacement fromequilibrium of a weight oscillating on the end of aspring is given by where isthe displacement (in feet) and is the time (in seconds).Use a graphing utility to graph the displacement function for Find the time beyond whichthe displacement does not exceed 1 foot from equilibrium.

101. Projectile Motion A batted baseball leaves the bat at an angle of with the horizontal and an initial velocityof feet per second. The ball is caught by anoutfielder 300 feet from home plate (see figure). Find

where the range of a projectile is given by


(a) Find the zero of in the interval

(b) A quadratic approximation of near is

Use a graphing utility to graph and in the sameviewing window. Describe the result.

(c) Use the Quadratic Formula to approximate thezeros of Compare the zero in the interval with the result of part (a).

�0, 6�g.

gf

g�x� � �0.45x2 � 5.52x � 13.70.

x � 4f

�0, 6�.f

f�x� � 3 sin�0.6x � 2�.

r = 300 ft Not drawn to scale

θ

r �1

32v0

2 sin 2�.

r�

v0 � 100�

0 � t � 10.

tyy � 1.56e�0.22t cos 4.9t,

Equilibrium

y

0 � t � 1.�y � 0�

ty

y �112�cos 8t � 3 sin 8t�

��8, 8�?�sin x)�x � 0

x → 0.

− π π −1

2

3

−2−3

x

y

f�x� �sin x

x

cos�1�x� � 0

��1, 1�?cos�1�x� � 0

x → 0.

x − π π

1

2

−2

y

f �x� � cos 1

x

t � 1t

U � 58.3 � 32.5 cos �t6

U

Vlad Turchenko 2010/used under license from Shutterstock.com

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103. Geometry The area of a rectangle inscribed in onearc of the graph of (see figure) is given by

(a) Use a graphing utility to graph the area function,and approximate the area of the largest inscribedrectangle.

(b) Determine the values of for which

Approximating the Number of Intersections InExercises 105 and 106, use the graph to approximate thenumber of points of intersection of the graphs of and

105. 106.


107. All trigonometric equations have either an infinitenumber of solutions or no solution.

108. The solutions of any trigonometric equation canalways be found from its solutions in the interval

109. If you correctly solve a trigonometric equation downto the statement then you can finish solvingthe equation by using an inverse trigonometric function.

110. The equation has three times thenumber of solutions in the interval as the equation

111. Writing Describe the difference between verifyingan identity and solving an equation.

Cumulative Mixed ReviewConverting from Degrees to Radians In Exercises113–116, convert the angle measure from degrees toradians. Round your answer to three decimal places.

113. 114.

115. 116.

117. Make a Decision To work an extended applicationanalyzing the normal daily high temperatures inPhoenix and in Seattle, visit this textbook’s CompanionWebsite. (Data Source: NOAA)

�210.55�0.41

486124

2 sin t � 1 � 0.�0, 2��

2 sin 3t � 1 � 0

sin x � 3.4,

�0, 2��.

x

2π

23

−3−4

4

y1

y2

y

x

2π

1234

y1

y2

y

y2 �12 x � 1y2 � 3x � 1

y1 � 2 sin xy1 � 2 sin x

y2.y1

A 1.x

2 −

−1

π 2 π x

x

yy = cos x

0 � x ��

2.A � 2x cos x,

y � cos x

104. MODELING DATAThe unemployment rates in the United States from2000 through 2008 are shown in the table, where represents the year, with corresponding to 2000.(Source: U.S. Bureau of Labor Statistics)

(a) Use a graphing utility to create a scatter plot of thedata.

(b) A model for the data is given by

Graph the model with the scatter plot from part (a).Is the model a good fit for the data? Explain.

(c) What term in the model gives the average unemployment rate? What is the rate?

(d) Economists study the lengths of business cycles,such as unemployment rates. Based on this shortspan of time, use the model to determine the lengthof this cycle.

(e) Use the model to estimate the next time the unemployment rate will be 5% or less.

r � 0.90 sin�1.04t � 1.62� � 5.23.

t � 0t

r

Year, t Rate, r

0 4.01 4.72 5.83 6.04 5.55 5.16 4.67 4.68 5.8

112. C A P S T O N E Consider the equation Explain the similarities and differences among finding all solutions in the interval findingall solutions in the interval and finding thegeneral solution.

�0, 2��,�0, ��2�,

2 sin x � 1.

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Section 5.4 Sum and Difference Formulas 377

Using Sum and Difference FormulasIn this section and the following section, you will study the uses of several trigonometricidentities and formulas.

Example 1 shows how sum and difference formulas can be used to find exact values of trigonometric functions involving sums or differences of special angles.

Example 1 Evaluating a Trigonometric Function

Find the exact value of (a) and (b)

Solutiona. Using the fact that with the formula for

yields

Try checking this result on your calculator. You will find that

b. Using the fact that

with the formula for yields

Now try Exercise 9.

��6 � �2

4.

��32 ��2

2 � �12��22 �

� sin �

3 cos

�

4� cos

�

3 sin

�

4

sin �

12� sin��

3�

�

4�sin�u � v�

�

12�

�

3�

�

4

cos 75� � 0.259.

��6 � �2

4.

��3

2 ��2

2 � �1

2��2

2 � � cos 30� cos 45� � sin 30� sin 45�

cos 75� � cos�30� � 45��

cos�u � v�75� � 30� � 45�

sin �

12.cos 75�

5.4 Sum and Difference Formulas

What you should learn● Use sum and difference formulas

to evaluate trigonometric

functions, verify trigonometric

identities, and solve trigonometric

equations.

Why you should learn itYou can use sum and difference

formulas to rewrite trigonometric

expressions. For instance, Exercise 89

on page 382 shows how to use sum

and difference formulas to rewrite a

trigonometric expression in a form

that helps you find the equation of a

standing wave.

Sum and Difference Formulas (See proofs on page 400.)

cos�u � v� � cos u cos v � sin u sin v


sin�u � v� � sin u cos v � cos u sin v


tan�u � v� �tan u � tan v

1 � tan u tan v


1 � tan u tan v

Konstantin Shevtsov 2010/ used under license from Shutterstock.com

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Example 2 Evaluating a Trigonometric Expression

Find the exact value of given

where and where

SolutionBecause and is in Quadrant I, as shown in Figure 5.19.Because and is in Quadrant II, as shown in Figure 5.20. You can find as follows.

Figure 5.19 Figure 5.20


Example 3 An Application of a Sum Formula

Write as an algebraic expression.

SolutionThis expression fits the formula for Angles

and

are shown in Figures 5.21 and 5.22, respectively.

Figure 5.21 Figure 5.22


x

1 − x21

v 1

1

u

2

�x � �1 � x2

�2.

�1

�2� x �

1

�2� �1 � x2

cos�u � v� � cos�arctan 1�cos�arccos x� � sin�arctan 1�sin�arccos x�

v � arccos xu � arctan 1

cos�u � v�.

cos�arctan 1 � arccos x�

v 13

12x

y

132 − 122 = 5

u

5

52 − 42 = 3

4

x

y

� �45��

1213� � �3

5��513� � �

4865

�1565

� �3365


sin�u � v�sin v � 5�13,vcos v � �12�13

cos u � 3�5,usin u � 4�5

�

2< v < �.cos v � �

1213

,0 < u <�

2sin u �

45

,

sin�u � v� Explore the ConceptUse a graphing utilityto graph

andin the same

viewing window. What can youconclude about the graphs? Is ittrue that

Use the graphing utility to graph

and

in the same viewing window.What can you conclude aboutthe graphs? Is it true that

sin�x � 4� � sin x � sin 4?

y2 � sin x � sin 4

y1 � sin�x � 4�

cos�x � 2� � cos x � cos 2?

y2 � cos x � cos 2y1 � cos�x � 2�

Andresr 2010/used under license from Shutterstock.com

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The next example shows how to use a difference formula to prove the cofunctionidentity

Example 4 Proving a Cofunction Identity

Prove the cofunction identity

SolutionUsing the formula for you have


Sum and difference formulas can be used to derive reduction formulas involvingexpressions such as

and

where is an integer.

Example 5 Deriving Reduction Formulas

Simplify each expression.

a.

b.

Solutiona. Using the formula for you have

b. Using the formula for you have

Note that the period of is so the period of is the same as theperiod of


tan �.tan�� 3��,tan �

� tan �.

�tan � � 0

1 � �tan ��0�

tan�� 3�� tan � � tan 3�

1 � tan � tan 3�

tan�u � v�,

� �sin �.

� �cos ��0� � �sin ��1�

cos�� 3�

2 � � cos � cos 3�

2� sin � sin

3�

2

cos�u � v�,

tan�� 3��

cos�� 3�

2 �

n

cos�� n�

2 �sin�� n�

2 �

� sin x.

� �0��cos x� � �1��sin x�

cos��

2� x� � cos

�

2 cos x � sin

�

2 sin x

cos�u � v�,

cos��

2� x� � sin x.

cos��

2� x� � sin x.

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The next example was taken from calculus. It is used to derive the formula for thederivative of the cosine function.

Example 7 An Application from Calculus

Verify that

where

SolutionUsing the formula for you have


� �cos x��cos h � 1

h � � �sin x��sin h

h �.

�cos x�cos h � 1�

h�

sin x sin hh

�cos x�cos h � 1� � sin x sin h

h

cos�x � h� � cos x

h�

cos x cos h � sin x sin h � cos x

h

cos�u � v�,

h 0.

cos�x � h� � cos x

h� �cos x�� cos h � 1

h � � �sin x��sin h

h �



in the interval 0, 2��.

sin�x ��

4� � sin�x ��

4� � �1

Algebraic SolutionUsing sum and difference formulas, rewrite the equation as

So, the only solutions in the interval are

and


x �7�

4.x �

5�

4

0, 2��

sin x � ��2

2.

sin x � �1�2

2�sin x��2

2 � � �1

2 sin x cos �

4� �1

sin x cos �

4� cos x sin

�

4� sin x cos

�

4� cos x sin

�

4� �1

Graphical Solution

Figure 5.23

From Figure 5.23, you can conclude that the approximate solutions in the interval are

and x � 5.498 �7�

4.x � 3.927 �

5�

4

0, 2��

−1

3

�20

y = sin x + + sin x − + 1π4 ( π

4( ((

The x-intercepts arex ≈ 3.927 and x ≈ 5.498.

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Evaluating a Trigonometric Expression In Exercises9–14, find the exact value of each expression.

9. (a) (b)

10. (a) (b)

11. (a) (b)

12. (a) (b)

13. (a) (b)

14. (a) (b)

Evaluating Trigonometric Functions In Exercises 15–30,find the exact values of the sine, cosine, and tangent of the angle.

15. 16.

17. 18.

19. 20.

21. 22.

23. 24.

25. 26.

27. 28.

29. 30.

Rewriting a Trigonometric Expression In Exercises31–38, write the expression as the sine, cosine, or tangentof an angle.

31.

32.

33. 34.

35.

36.

37.

38.

Evaluating a Trigonometric Expression In Exercises39–44, find the exact value of the expression.

39.

40.

41.

42.

43. 44.

Algebraic-Graphical-Numerical In Exercises 45–48, usea graphing utility to complete the table and graph thetwo functions in the same viewing window. Use both thetable and the graph as evidence that Then verifythe identity algebraically.

45.

46.

47.

48. y2 � sin2 xy1 � sin�x � �� sin�x � ��,y2 � cos2 xy1 � cos�x � �� cos�x � ��,

y2 � ��2

2�cos x � sin x�y1 � cos�5�

4� x�,

y2 �1

2�cos x � �3 sin x�y1 � sin��

6� x�,

y1 � y2.

tan 25� � tan 110�

1 � tan 25� tan 110�

tan�5��6� � tan��6�1 � tan�5��6� tan��6�

cos 120� cos 30� � sin 120� sin 30�

sin 120� cos 60� � cos 120� sin 60�

cos �

16 cos

3�

16� sin

�

16 sin

3�

16

sin �

12 cos

�

4� cos

�

12 sin

�

4

sin 4�

9 cos

�

8� cos

4�

9 sin

�

8

cos �

9 cos

�

7� sin

�

9 sin

�

7

cos 0.96 cos 0.42 � sin 0.96 sin 0.42

sin 3.5 cos 1.2 � cos 3.5 sin 1.2

tan 154� � tan 49�

1 � tan 154� tan 49�

tan 325� � tan 116�

1 � tan 325� tan 116�

sin 110� cos 80� � cos 110� sin 80�

cos 60� cos 10� � sin 60� sin 10�

�13�

12�

7�

12

5�

1213�

12

�165��285�

15�75�

�19�

12�

2�

3�

9�

4�

�

12�

�

6�

�

4

17�

12�

7�

6�

�

4

11�

12�

3�

4�

�

6

255� � 300� � 45�195� � 225� � 30�

165� � 135� � 30�105� � 60� � 45�

sin 7�

6� sin

�

3sin�7�

6�

�

3�sin 135� � sin 30�sin�135� � 30��

sin 2�

3� sin

5�

6sin�2�

3�

5�

6 �

cos �

4� cos

�

3cos��

4�

�

3�sin 405� � sin 120�sin�405� � 120��cos 240� � cos 0�cos�240� � 0��

Vocabulary and Concept CheckIn Exercises 1–6, fill in the blank to complete the trigonometric formula.

1. _______ 2. _______ 3. _______

4. _______ 5. _______ 6. _______

7. Rewrite so that you can use a sum formula.

8. Rewrite so that you can use a difference formula.


cos �

12

sin 105�

tan�u � v� �cos�u � v� �sin�u � v� �

tan�u � v� �cos�u � v� �sin�u � v� �


x 0.2 0.4 0.6 0.8 1.0 1.2 1.4

y1

y2

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Evaluating a Trigonometric Expression In Exercises49–52, find the exact value of the trigonometricexpression given that and (Both and are in Quadrant II.)

49. 50.

51. 52.

Evaluating a Trigonometric Expression In Exercises53–56, find the exact value of the trigonometricexpression given that and (Both and are in Quadrant III.)

53. 54.

55. 56.

An Application of a Sum or Difference Formula InExercises 57–60, write the trigonometric expression asan algebraic expression.

57. 58.

59.

60.

Evaluating a Trigonometric Expression In Exercises61–68, find the value of the expression without using acalculator.

61. 62.

63. 64.

65. 66.

67. 68.

Evaluating a Trigonometric Expression In Exercises69–72, use right triangles to evaluate the expression.

69.

70.

71. 72.


73. 74.

75.

76.

77.

78.

79.

80.

Solving a Trigonometric Equation In Exercises 81–84,find the solution(s) of the equation in the interval Use a graphing utility to verify your results.

81.

82.

83.

84.

Approximating Solutions Graphically In Exercises85–88, use a graphing utility to approximate the solutions of the equation in the interval

85.

86.

87. 88.

89. (p. 377) The equation of astanding wave is obtained by adding the displacements of two waves traveling inopposite directions (see figure). Assumethat each of the waves has amplitude period and wavelength The models fortwo such waves are

and

Show that

t = 0

t T = 18

t T= 28

y1

y1

y1

y2

y2

y2

y1 + y2

y1 + y2

y1 + y2

y1 � y2 � 2A cos 2�t

T cos

2�x

.

y2 � A cos 2�� t

T�

x

�.y1 � A cos 2�� t

T�

x

�

.T,A,

cos�x ��

2� � sin2 xsin�x ��

2� � �cos2 x

sin�x ��

2� � cos�x �3�

2 � � 0

cos�x ��

4� � cos�x ��

4� � 1

[0, 2�.

2 sin�x ��

2� � 3 tan�� x� � 0

tan�x � �� 2 sin�x � �� 0

cos�x ��

6� � cos�x ��

6� � 1

sin�x ��

3� � sin�x ��

3� � 1

[0, 2�.

sin�x � y� sin�x � y� � sin2 x � sin2 y

cos�x � y� cos�x � y� � cos2 x � sin2 y

cos�x � y� � cos�x � y� � 2 cos x cos y

sin�x � y� � sin�x � y� � 2 sin x cos y

tan��

4� ��

1 � tan �

1 � tan �

tan�x � �� tan�� x� � 2 tan x

sin�3� � x� � sin xsin��

2� x� � cos x

tan�sin�1 45

� cos�1 513�sin�tan�1

34

� sin�1 35�

cos�sin�1 1213

� cos�1 817�

sin�cos�1 35

� sin�1 513�

cos�cos�1 0 � cos�1 1�sin�sin�1 0 � cos�1 0�cos�sin�1 1 � cos�1 0�sin�sin�1 1 � cos�1 1�cos� � cos�1��1��cos�� sin�1 1�

sincos�1��1� � ��sin��

2� sin�1��1�

cos�arcsin x � arctan 2x�sin�arctan 2x � arccos x�

cos�arccos x � arcsin x�sin�arcsin x � arccos x�

cos�u � v�sin�v � u�tan�u � v�cos�u � v�

vucos v � �3

5.sin u � � 817

sin�u � v�tan�u � v�cos�v � u�sin�u � v�

vucos v � �4

5.sin u � 513


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90. Harmonic Motion A weight is attached to a spring suspended vertically from a ceiling. When a drivingforce is applied to the system, the weight moves vertically from its equilibrium position, and thismotion is modeled by

where is the distance from equilibrium (in feet) andis the time (in seconds).

(a) Use a graphing utility to graph the model.

(b) Use the identity

where to write the modelin the form Use thegraphing utility to verify your result.

(c) Find the amplitude of the oscillations of the weight.

(d) Find the frequency of the oscillations of the weight.

ConclusionsTrue or False? In Exercises 91 and 92, determinewhether the statement is true or false. Justify your answer.

91.

92.

Verifying Identities In Exercises 93–96, verify the identity.

93. is an integer.

94. is an integer.

95. whereand

96.where and

Rewriting a Trigonometric Expression In Exercises97–100, use the formulas given in Exercises 95 and 96 to write the expression in the following forms. Use agraphing utility to verify your results.

(a) (b)

97. 98.

99. 100.

Rewriting a Trigonometric Expression In Exercises 101and 102, use the formulas given in Exercises 95 and 96 to write the trigonometric expression in the form

101. 102.

103. An Application from Calculus Verify the followingidentity used in calculus.

104. Exploration Let in the identity in Exercise103 and define the functions and as follows.

(a) What are the domains of the functions and

(b) Use a graphing utility to complete the table.

(c) Use the graphing utility to graph the functions and

(d) Use the table and graph to make a conjectureabout the values of the functions and as

105. Proof Three squares of side are placed side by side(see figure). Make a conjecture about the relationshipbetween the sum and Prove your conjectureby using the formula for

107. Write a sum formula for

108. Write a sum formula for

Cumulative Mixed ReviewFinding Intercepts In Exercises 109–112, find the - and-intercepts of the graph of the equation. Use a graphing

utility to verify your results.

109. 110.

111. 112. y � 2x�x � 7y � �2x � 9� � 5

y � x2 � 3x � 40y � �12�x � 10� � 14

yx

tan�u � v � w�.cos�u � v � w�.

s

s

u v w s s

tan�u � v�.w.u � v

s

h → 0.gf

g.f

g?f

g�h� � cos �

3�sin h

h � � sin �

3�1 � cos h

h �

f�h� �sin��3 � h� � sin��3�

h

gfx � ��3

sin�x � h� � sin x

h�

cos x sin h

h�

sin x�1 � cos h�h

5 cos��

4�2 sin��

2�a sin B� � b cos B�.

sin 2� � cos 2�12 sin 3� � 5 cos 3�

3 sin 2� � 4 cos 2�sin � � cos �

�a2 � b2 cos�B� � C�a2 � b2 sin�B� � C

b > 0.C � arctan�a�b�a sin B� � b cos B� ��a2 � b2 cos�B� � C�,

a > 0.C � arctan�b�a�a sin B� � b cos B� ��a2 � b2 sin�B� � C�,

nsin�n� � �� 1�n sin �,

ncos�n� � �� 1�n cos �,

sin�x �11�

2 � � cos x

cos�u ± v� � cos u ± cos v

y � �a2 � b2 sin�Bt � C�.a > 0,C � arctan�b�a�,

a sin B� � b cos B� � �a2 � b2 sin�B� � C�

ty

y �1

3 sin 2t �

1

4 cos 2t

h 0.01 0.02 0.05 0.1 0.2 0.5

f�h�

g�h�

106. C A P S T O N E Give an example to justify eachstatement.

(a)

(b)

(c) tan�u � v� tan u � tan v

cos�u � v� cos u � cos v

sin�u � v� sin u � sin v

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Multiple-Angle FormulasIn this section, you will study four additional categories of trigonometric identities.

1. The first category involves functions of multiple angles such as

and

2. The second category involves squares of trigonometric functions such as

3. The third category involves functions of half-angles such as

4. The fourth category involves products of trigonometric functions such as

You should learn the double-angle formulas below because they are used often in trigonometry and calculus.

Example 1 Solving a Multiple-Angle Equation

Solve

SolutionBegin by rewriting the equation so that it involves functions of (rather than Thenfactor and solve as usual.


Double-angle formula

Factor.

Set factors equal to zero.

Isolate trigonometric functions.

Solutions in

So, the general solution is


where is an integer. Try verifying this solution graphically.


n

x �3�

2� 2n�x �

�

2� 2n�

�0, 2�� x �3�

2 x �

�

2,

3�

2

sin x � �1 cos x � 0

1 � sin x � 0 2 cos x � 0

2 cos x�1 � sin x� � 0

2 cos x � 2 sin x cos x � 0

2 cos x � sin 2x � 0

2x).x

2 cos x � sin 2x � 0.

sin u cos v.

sin u

2.

sin2 u.

cos ku.sin ku


5.5 Multiple-Angle and Product-to-Sum Formulas

What you should learn● Use multiple-angle formulas

to rewrite and evaluate

trigonometric functions.

● Use power-reducing formulas



● Use half-angle formulas



● Use product-to-sum and

sum-to-product formulas



Why you should learn itYou can use a variety of trigonometric

formulas to rewrite trigonometric

functions in more convenient forms.

For instance, Exercise 126 on page

393 shows you how to use a half-

angle formula to determine the apex

angle of a sound wave cone caused

by the speed of an airplane.

tan 2u �2 tan u

1 � tan2 u

Double-Angle Formulas (See the proofs on page 401.)

sin 2u � 2 sin u cos u

cos 2u

�

�

�

cos2 u � sin2 u

2 cos2 u � 1

1 � 2 sin2 u

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Section 5.5 Multiple-Angle and Product-to-Sum Formulas 385

Example 2 Evaluating Functions Involving Double Angles

Use the following to find and

SolutionIn Figure 5.24, you can see that

and

Consequently, using each of the double-angle formulas, you can write the double anglesas follows.


The double-angle formulas are not restricted to the angles and Other doublecombinations, such as and or and are also valid. Here are two examples.

and

By using double-angle formulas together with the sum formulas derived in the precedingsection, you can form other multiple-angle formulas.

Example 3 Deriving a Triple-Angle Formula

Rewrite in terms of

SolutionRewrite as a sum.

Sum formula

Double-angle formula

Multiply.


Multiply.

Simplify.


� 3 sin x � 4 sin3 x

� 2 sin x � 2 sin3 x � sin x � 2 sin3 x

� 2 sin x�1 � sin2 x� � sin x � 2 sin3 x

� 2 sin x cos2 x � sin x � 2 sin3 x

� 2 sin x cos x cos x � �1 � 2 sin2 x� sin x

� sin 2x cos x � cos 2x sin x

sin 3x � sin�2x � x�

sin x.sin 3x

cos 6� � cos2 3� � sin2 3�sin 4� � 2 sin 2� cos 2�

3�,6�2�4��.2�

tan 2� �2 tan �

1 � tan2 ��

2��12�5�1 � ��12�5�2 �

120119

� �119169

� �120169

� 2� 25

169� � 1 � 2��1213��

513�

cos 2� � 2 cos2 � � 1 sin 2� � 2 sin � cos �

tan � � �125

.

sin � �yr

� �1213

3�

2< � < 2�cos � �

5

13,

tan 2�.cos 2�,sin 2�,

(5, −12)

6 4 2 −2

−2

−4

−4

−6

−8

−10

−12

13

θ x

y

Figure 5.24

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Power-Reducing FormulasThe double-angle formulas can be used to obtain the following power-reducing formulas.

Example 4 shows a typical power reduction that is used in calculus. Note therepeated use of power-reducing formulas.

Example 4 Reducing a Power

Rewrite as a sum of first powers of the cosines of multiple angles.

Solution

Property of exponents

Power-reducing formula

Expand binomial.

Power-reducing formula


Simplify.

Factor.

You can use a graphing utility to check this result, as shown in Figure 5.25. Notice thatthe graphs coincide.

Figure 5.25


−2

−� �

2

y2 = (3 − 4 cos 2x + cos 4x)18

y1 = sin4 x

�1

8�3 � 4 cos 2x � cos 4x�

�3

8�

1

2 cos 2x �

1

8 cos 4x

�1

4�

1

2 cos 2x �

1

8�

1

8 cos 4x

�1

4�1 � 2 cos 2x �1 � cos 4x

2 �

�1

4�1 � 2 cos 2x � cos2 2x�

� �1 � cos 2x

2 �2

sin4 x � �sin2 x�2

sin4 x

Power-Reducing Formulas (See the proofs on page 401.)

tan2 u �1 � cos 2u

1 � cos 2u

cos2 u �1 � cos 2u

2

sin2 u �1 � cos 2u

2

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Half-Angle FormulasYou can derive some useful alternative forms of the power-reducing formulas by replacing

with The results are called half-angle formulas.

Example 5 Using a Half-Angle Formula

Find the exact value of

SolutionBegin by noting that 105 is half of 210 . Then, using the half-angle formula for

and the fact that 105 lies in Quadrant II, you have

The positive square root is chosen because is positive in Quadrant II.


sin �

��1 � ��3�2�2

��2 � �3

2.

��1 � ��cos 30��2

sin 105� ��1 � cos 210�

2

�sin�u�2��

sin 105�.

u�2.u

Half-Angle Formulas

The signs of and depend on the quadrant in which lies.u2

cos u2

sin u2

tan u

2�

1 � cos u

sin u�

sin u

1 � cos u

cos u

2� ±�1 � cos u

2sin

u

2� ±�1 � cos u

2

Technology TipUse your calculator to verify the resultobtained in Example 5.

That is, evaluate and

You will noticethat both expressions yield thesame result.

��2 � �3 ��2.

sin 105�


Find all solutions of in the interval �0, 2��.1 � cos2 x � 2 cos2 x2

Algebraic Solution


Half-angle formula

Simplify.

Simplify.

Factor.

By setting the factors and equal to zero, you find thatthe solutions in the interval are and


x � 0.x � 3��2,x � ��2,�0, 2��

cos x � 1cos x

cos x�cos x � 1� � 0

cos2 x � cos x � 0

1 � cos2 x � 1 � cos x

1 � cos2 x � 2�±�1 � cos x2

�2

1 � cos2 x � 2 cos2 x2

Graphical Solution

Figure 5.26

From Figure 5.26, you can conclude that the approximate

solutions of in the interval

are and x 4.712 3�

2.x 1.571

�

2,x � 0,

�0, 2��1 � cos2 x � 2 cos2 x2

−1

3 y = 1 + cos2 x − 2 cos2 x2

�2�2

−

The x-intercepts arex = 0, x ≈ 1.571, andx ≈ 4.712.

Study TipTo find the exact valueof a trigonometricfunction with an angle

in form using a half-angle formula, first convert the angle measure to decimaldegree form. Then multiply theangle measure by 2.

D�M�S

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Product-to-Sum FormulasEach of the following product-to-sum formulas is easily verified using the sum anddifference formulas discussed in the preceding section.

Product-to-sum formulas are used in calculus to evaluate integrals involving theproducts of sines and cosines of two different angles.

Example 7 Writing Products as Sums

Rewrite the product as a sum or difference.

SolutionUsing the appropriate product-to-sum formula, you obtain


Occasionally, it is useful to reverse the procedure and write a sum of trigonometricfunctions as a product. This can be accomplished with the following sum-to-productformulas.

�1

2 sin 9x �

1

2 sin x.

cos 5x sin 4x �1

2�sin�5x � 4x� � sin�5x � 4x�

cos 5x sin 4x

Product-to-Sum Formulas

cos u sin v �1

2�sin�u � v� � sin�u � v�

sin u cos v �1

2�sin�u � v� � sin�u � v�

cos u cos v �1

2�cos�u � v� � cos�u � v�

sin u sin v �1

2�cos�u � v� � cos�u � v�

Technology TipYou can use a graphingutility to verify thesolution in Example 7.

Graph andin the

same viewing window. Noticethat the graphs coincide. So,you can conclude that the twoexpressions are equivalent.

y2 �12 sin 9x �

12 sin x

y1 � cos 5x sin 4x

Sum-to-Product Formulas (See proof on page 402.)

cos u � cos v � �2 sin�u � v

2 � sin�u � v

2 �

cos u � cos v � 2 cos�u � v

2 � cos�u � v

2 �

sin u � sin v � 2 cos�u � v

2 � sin�u � v

2 �

sin u � sin v � 2 sin�u � v

2 � cos�u � v

2 �

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Example 8 Using a Sum-to-Product Formula

Find the exact value of

SolutionUsing the appropriate sum-to-product formula, you obtain




in the interval

Solution


Sum-to-product formula

Simplify.

By setting the factor equal to zero, you can find that the solutions in the intervalare

The equation yields no additional solutions. You can use a graphing utility toconfirm the solutions, as shown in Figure 5.27.

Figure 5.27

Notice that the general solution is

where is an integer.


n

x �n�

4

−3

3

�2�4

−

y = sin 5x + sin 3x

cos x � 0

x � 0, �

4,

�

2,

3�

4, �,

5�

4,

3�

2,

7�

4.

�0, 2��sin 4x

2 sin 4x cos x � 0

2 sin�5x � 3x

2 � cos�5x � 3x

2 � � 0

sin 5x � sin 3x � 0

�0, 2��.

sin 5x � sin 3x � 0

� ��6

2.

� 2��3

2 ��2

2 � � 2 cos 150� cos 45�

cos 195� � cos 105� � 2 cos�195� � 105�

2 � cos�195� � 105�

2 �

cos 195� � cos 105�.

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Finding Exact Values of Trigonometric Functions InExercises 9 and 10, use the figure to find the exact valueof each trigonometric function.

9.

(a) (b)

(c) (d)

(e) (f)

(g) (h)

10.

(a) (b)

(c) (d)

(e) (f)

(g) (h)

Solving a Multiple-Angle Equation In Exercises 11–20,use a graphing utility to approximate the solutions of theequation in the interval If possible, find the exactsolutions algebraically.

11. 12.

13. 14.

15. 16.

17. 18.

19. 20.

Evaluating Functions Involving Double Angles InExercises 21–26, find the exact values of and using the double-angle formulas.

21.

22.

23.

24.

25.

26.

Using a Double-Angle Formula In Exercises 27–30, usea double-angle formula to rewrite the expression. Use agraphing utility to graph both expressions to verify thatboth forms are the same.

27. 28.

29.

30.

Reducing a Power In Exercises 31–44, rewrite theexpression in terms of the first power of the cosine. Usea graphing utility to graph both expressions to verifythat both forms are the same.

31. 32.

33. 34.

35. 36.

37. 38.

39. 40.

41. 42.

43. 44. cos4 x2

sin4 x2

sin2 x tan2 xtan2 2x

sin2 x2

cos2 x2

cos2 2xsin2 2x

sin4 x cos2 xsin2 x cos4 x

cos6 xsin2 x cos2 x

sin8 xcos4 x

�cos x � sin x��cos x � sin x�4 � 8 sin2 x

4 sin x cos x � 18 sin x cos x

��2 < u < �csc u � 3,

��2 < u < �sec u � �2,

3��2 < u < 2�cot u � �6,

� < u < 3��2tan u �12,

��2 < u < �cos u � �23,

0 < u < ��2sin u �35,

tan 2ucos 2u,sin 2u,

tan 2x � 2 cos x � 0cos 2x � sin x � 0

�sin 2x � cos 2x�2 � 1sin 4x � �2 sin 2x

tan 2x � cot x � 0cos 2x � cos x � 0

sin 2x sin x � cos x4 sin x cos x � 1

sin 2x � cos x � 0sin 2x � sin x � 0

[0, 2��.

csc 2�sec 2�

cot 2�tan 2�

cos 2�sin 2�

tan �sin �

5

12

θ

cot 2�csc 2�

sec 2�cos 2�

sin 2�tan �

cos �sin �

3

4 θ

Vocabulary and Concept CheckIn Exercises 1–6, fill in the blank to complete the trigonometric formula.

1. _______ 2. _______ 3. _______

4. _______ 5. _______ 6. _______

7. Match each function with its double-angle formula. 8. Match each expression with its product-to-sum formula.

(a) (i) (a) (i)

(b) (ii) (b) (ii)

(c) (iii)(c) (iii)


12�sin�u � v� � sin�u � v�cos u cos v2 tan u

1 � tan2 utan 2u

12�sin�u � v� � sin�u � v�cos u sin v2 sin u cos ucos 2u

12�cos�u � v� � cos�u � v�sin u cos v1 � 2 sin2 usin 2u

�1 � cos 2u1 � cos 2u

�sin u

1 � cos usin

u2

�

cos u � cos v �sin u sin v �cos2 u �


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Finding Exact Values of Trigonometric Functions InExercises 45 and 46, use the figure to find the exact valueof each trigonometric function.

45.

(a) (b)

(c) (d)

(e) (f)

(g) (h)

46.

(a) (b)

(c) (d)

(e) (f)

(g) (h)

Using a Half-Angle Formula In Exercises 47–54, use thehalf-angle formulas to determine the exact values of thesine, cosine, and tangent of the angle.

47. 48.

49. 50.

51. 52.

53. 54.

Using a Half-Angle Formula In Exercises 55–60, findthe exact values of and usingthe half-angle formulas.

55.

56.

57.

58.

59.

60.

Using a Half-Angle Formula In Exercises 61–64, use thehalf-angle formulas to simplify the expression.

61. 62.

63. 64.

Solving a Trigonometric Equation In Exercises 65–68,find the solutions of the equation in the interval Use a graphing utility to verify your answers.

65. 66.

67. 68.

Writing Products as Sums In Exercises 69–76, use theproduct-to-sum formulas to write the product as a sumor difference.

69. 70.

71. 72.

73. 74.

75. 76.

Writing Sums as Products In Exercises 77–84, use thesum-to-product formulas to write the sum or differenceas a product.

77. 78.

79. 80.

81.

82.

83.

84.

Using a Sum-to-Product Formula In Exercises 85–88,use the sum-to-product formulas to find the exact valueof the expression.

85. 86.

87. 88.

Solving a Trigonometric Equation In Exercises 89–92,find the solutions of the equation in the interval Use a graphing utility to verify your answers.

89. 90.

91. 92. sin2 3x � sin2 x � 0cos 2x

sin 3x � sin x� 1 � 0

cos 2x � cos 6x � 0sin 6x � sin 2x � 0

[0, 2��.

sin 5�

4� sin

3�

4cos

5�

12� cos

�

12

cos 120� � cos 30�sin 195� � sin 105�

sin�x ��

2� � sin�x ��

2�

cos��

2� � cos��

2�cos� � 2�� cos

sin�� sin�� sin x � sin 5xcos 6x � cos 2x

sin 3� � sin �sin 5� � sin �

sin�x � y� cos�x � y�sin�x � y� sin�x � y�6 sin 45� cos 15�10 cos 75� cos 15�

3 sin 2� sin 3�sin 5� cos 3�

4 cos �

3 sin

5�

66 sin

�

3 cos

�

3

tan x

2� sin x � 0cos

x

2� sin x � 0

sin x

2� cos x � 1 � 0sin

x

2� cos x � 0

[0, 2��.

��1 � cos�x � 1�2

��1 � cos 8x

1 � cos 8x

�1 � cos 4x

2�1 � cos 6x

2

��2 < u < �sec u � �72,

� < u < 3��2csc u � �53,

� < u < 3��2cot u � 3,

3��2 < u < 2�tan u � �85,

0 < u < ��2cos u �35,

��2 < u < �sin u �513,

tan�u/2�cos�u/2�,sin�u/2�,

7�

123�

8

�

12

�

8

157� 30�67� 30�

165�75�

2 cos �

2 tan

�

22 sin

�

2 cos

�

2

csc �

2sec

�

2

cot �

2tan

�

2

cos �

2sin

�

2

7

24

θ

2 cos �

2 tan

�

22 sin

�

2 cos

�

2

cot �

2csc

�

2

sec �

2tan

�

2

sin �

2cos

�

2

8

15 θ

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Using Trigonometric Identities In Exercises 93–96, usethe figure and trigonometric identities to find the exactvalue of the trigonometric function in two ways.

93. 94.

95. 96.

Verifying Trigonometric Identities In Exercises 97–110,verify the identity algebraically. Use a graphing utility tocheck your result graphically.

97. 98.

99.

100.

101.

102.

103.

104.

105.

106.

107.

108.

109.

110.

Using a Power-Reducing Formula In Exercises 111–114,rewrite the function using the power-reducing formulas.Then use a graphing utility to graph the function.

111. 112.

113. 114.

Rewriting a Trigonometric Expression In Exercises115–120, write the trigonometric expression as an algebraic expression.

115. 116.

117. 118.

119. 120.

An Application from Calculus In Exercises 121 and 122,the graph of a function is shown over the interval

(a) Find the -intercepts of the graph of algebraically. Verify your solutions by using the zero orroot feature of a graphing utility. (b) The -coordinatesof the extrema of are solutions of the trigonometricequation (calculus is required to find the trigonometricequation). Find the solutions of the equation algebraically.Verify these solutions using the maximum and minimumfeatures of the graphing utility.

121. Function:

Trigonometric equation:

122. Function:

Trigonometric equation:

123. Projectile Motion The range of a projectile fired atan angle with the horizontal and with an initialvelocity of feet per second is given by

where is measured in feet. An athlete throws a javelinat 75 feet per second. At what angle must the athletethrow the javelin so that the javelin travels 130 feet?

124. Geometry The length of each of the two equal sidesof an isosceles triangle is 10 meters (see figure). Theangle between the two sides is

(a) Write the area of the triangle as a function of

(b) Write the area of the triangle as a function of anddetermine the value of such that the area is a maximum.

��

��2.

θ 10 m10 m

�.

r

r �1

32 v0

2 sin 2�

v0

�

3

0

−3

2�

f

�2 sin 2x � cos x � 0

f�x� � cos 2x � sin x

3

0

−3

2�

f

2 cos 2x � cos x � 0

f�x� � sin 2x � sin x

fx

fx[0, 2�].f

sin�2 arctan x�cos�2 arctan x�sin�2 arccos x�cos�2 arcsin x�cos�2 arccos x�sin�2 arcsin x�

f�x� � sin3 x f �x� � cos4 x

f �x� � cos2 x f �x� � sin2 x

sin x � sin ycos x � cos y

� �cot x � y

2

sin x ± sin ycos x � cos y

� tan x ± y

2

sin 4� � 4 sin � cos ��1 � 2 sin2 ��cos 3� � cos3 � � 3 sin2 � cos �

tan u

2� csc u � cot u

sec u

2� ±� 2 tan u

tan u � sin u

cos 3�

cos �� 1 � 4 sin2 �

1 � cos 10y � 2 cos2 5y

sin �

3 cos

�

3�

12

sin 2�

3

�sin x � cos x�2 � 1 � sin 2x

cos4 x � sin4 x � cos 2x

cos2 2� � sin2 2� � cos 4�

sec 2� �sec2 �

2 � sec2 �csc 2� �

csc �2 cos �

cos � sin �sin � cos �

cos2 �sin2 �

3

4

12

5

αβ

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125. Mechanical Engineering When two railroad tracksmerge, the overlapping portions of the tracks are in theshape of a circular arc (see figure). The radius of eacharc (in feet) and the angle are related by

Write a formula for in terms of cos

126. (p. 384) The machnumber of an airplane is the ratio of itsspeed to the speed of sound. When an airplane travels faster than the speed ofsound, the sound waves form a cone behindthe airplane (see figure). The mach numberis related to the apex angle of the cone by

(a) Find the angle that corresponds to a mach number of 1.

(b) Find the angle that corresponds to a mach number of 4.5.

(c) The speed of sound is about 760 miles per hour.Determine the speed of an object having the machnumbers in parts (a) and (b).

(d) Rewrite the equation as a trigonometric function of

ConclusionsTrue or False? In Exercises 127 and 128, determinewhether the statement is true or false. Justify your answer.

127.

128. The graph of has a maximum at

129. Think About It Consider the function

(a) Use a graphing utility to graph the function.

(b) Make a conjecture about a function that is an identity with

(c) Verify your conjecture algebraically.

Cumulative Mixed ReviewFinding the Midpoint of a Line Segment In Exercises131–134, (a) plot the points, (b) find the distance betweenthe points, and (c) find the midpoint of the line segmentconnecting the points.

131. 132.

133. 134.

Finding the Complement and Supplement of Angles InExercises 135–138, find (if possible) the complement andsupplement of each angle.

135. (a) 55 (b) 162

136. (a) 109 (b) 78

137. (a) (b)

138. (a) 0.95 (b) 2.76

139. Find the radian measure of the central angle of a circlewith a radius of 15 inches that intercepts an arc oflength 7 inches.

140. Find the length of the arc on a circle of radius 21 centimeters intercepted by a central angle of 35�.

9�

20�

18

��

��

�13, 23�, ��1, �3

2��0, 12�, �43, 52�

��4, �3�, �6, 10��5, 2�, ��1, 4�

f.

f�x� � 2 sin x�2 cos2 x

2� 1�.

��, 4�.y � 4 � 8 sin2 x

� x 2�sin x2

� ��1 � cos x2

,

�.

�

�

θ

sin �

2�

1M

.

�

M

θ θ

r r

x

�.x

x2

� 2r sin2 �

2.

�r

130. C A P S T O N E Consider the function

(a) Use the power-reducing formulas to write thefunction in terms of cosine to the first power.

(b) Determine another way of rewriting the function.Use a graphing utility to rule out incorrectlyrewritten functions.

(c) Add a trigonometric term to the function so thatit becomes a perfect square trinomial. Rewritethe function as a perfect square trinomial minusthe term that you added. Use the graphing utilityto rule out incorrectly rewritten functions.

(d) Rewrite the result of part (c) in terms of the sineof a double angle. Use the graphing utility to ruleout incorrectly rewritten functions.

(e) When you rewrite a trigonometric expression, theresult may not be the same as a friend’s. Doesthis mean that one of you is wrong? Explain.

f�x� � sin4 x � cos4 x.

DCWcreations 2010/used under license from Shutterstock.com

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5 Chapter Summary

What did you learn? Explanation and Examples Review Exercises

5.1

Recognize and write the fundamental trigonometric identities (p. 350).

Reciprocal Identities

Quotient Identities

Pythagorean Identities

Cofunction Identities

Even/Odd Identities

cot��u� � �cot usec��u� � sec ucsc��u� � �csc u

tan��u� � �tan ucos��u� � cos usin��u� � �sin u

csc��2� � u� � sec usec��2� � u� � csc u

cot��2� � u� � tan utan��2� � u� � cot u

cos��2� � u� � sin usin��2� � u� � cos u

1 � cot2 u � csc2 u1 � tan2 u � sec2 usin2 u � cos2 u � 1

cot u �cos usin u

tan u �sin ucos u

cot u � 1�tan usec u � 1�cos ucsc u � 1�sin u

tan u � 1�cot ucos u � 1�sec usin u � 1�csc u

1–10

Use the fundamental trigonometricidentities to evaluate trigonometricfunctions, and simplify and rewritetrigonometric expressions (p. 351).

On occasion, factoring or simplifying trigonometric expressions can best be done by first rewriting the expression in terms of just one trigonometric function or in terms of sine or cosine alone.

11–26

5.2

Verify trigonometric identities (p. 357).

Guidelines for Verifying Trigonometric Identities

1. Work with one side of the equation at a time.

2. Look to factor an expression, add fractions, square abinomial, or create a monomial denominator.

3. Look to use the fundamental identities. Note which functions are in the final expression you want. Sines and cosines pair up well, as do secants and tangents,and cosecants and cotangents.

4. When the preceding guidelines do not help, try convertingall terms to sines and cosines.

5. Always try something.

27–38

5.3

Use standard algebraic techniquesto solve trigonometric equations(p. 365).

Use standard algebraic techniques such as collecting liketerms, extracting square roots, and factoring to solvetrigonometric equations.

39–50

Solve trigonometric equations ofquadratic type (p. 368).

To solve trigonometric equations of quadratic typefactor the quadratic or, when this

is not possible, use the Quadratic Formula.ax2 � bx � c � 0, 51–54

Solve trigonometric equationsinvolving multiple angles (p. 370).

To solve equations that contain forms such as orfirst solve the equation for then divide your result

by k.ku,cos ku,

sin ku55–62

Use inverse trigonometric functionsto solve trigonometric equations(p. 371).

After factoring an equation and setting the factors equal to0, you may get an equation such as In thiscase, use inverse trigonometric functions to solve. (SeeExample 10.)

tan x � 3 � 0.63–66

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Chapter Summary 395

What did you learn? Explanation and Examples Review Exercises

5.4

Use sum and difference formulas toevaluate trigonometric functions,verify trigonometric identities,and solve trigonometric equations(p. 377).

Sum and Difference Formulas


1 � tan u tan v


1 � tan u tan v





67–88

5.5

Use multiple-angle formulas torewrite and evaluate trigonometricfunctions (p. 384).

Double-Angle Formulas

tan 2u �2 tan u

1 � tan2 u

cos 2u � cos2 u � sin2 usin 2u � 2 sin u cos u89–98

Use power-reducing formulas torewrite and evaluate trigonometricfunctions (p. 386).

Power-Reducing Formulas

tan2 u �1 � cos 2u1 � cos 2u

cos2 u �1 � cos 2u

2


299–104

Use half-angle formulas to rewrite and evaluate trigonometricfunctions (p. 387).

Half-Angle Formulas

The signs of and depend on the quadrant

in which lies.u2

cos u2

sin u2

tan u2

�1 � cos u

sin u�

sin u1 � cos u

cos u2

� ±�1 � cos u2

sin u2

� ±�1 � cos u2

105–118

Use product-to-sum formulas and sum-to-product formulas torewrite and evaluate trigonometricfunctions (p. 388).

Product-to-Sum Formulas

Sum-to-Product Formulas

cos u � cos v � �2 sin�u � v2 sin�u � v

2

cos u � cos v � 2 cos�u � v2 cos�u � v

2

sin u � sin v � 2 cos�u � v2 sin�u � v

2

sin u � sin v � 2 sin�u � v2 cos�u � v

2

cos u sin v � �1�2� �sin�u � v� � sin�u � v��

sin u cos v � �1�2� �sin�u � v� � sin�u � v��

cos u cos v � �1�2� �cos�u � v� � cos�u � v��

sin u sin v � �1�2� �cos�u � v� � cos�u � v��

119–130

� 1 � 2 sin2 u

� 2 cos2 u � 1

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5 Review Exercises See www.CalcChat.com for worked-out solutions to odd-numbered exercises.For instructions on how to use a graphing utility, see Appendix A.

Recognizing Fundamental Trigonometric Identities InExercises 1–10, name the trigonometric function that isequivalent to the expression.

1. 2.

3. 4.

5. 6.

7. 8.

9. 10.

Using Identities to Evaluate a Function In Exercises11–14, use the given values to evaluate (if possible) theremaining trigonometric functions of the angle.

11.

12.

13.

14.

Simplifying a Trigonometric Expression In Exercises 15–24, use the fundamental identities to simplify theexpression. Use the table feature of a graphing utility tocheck your result numerically.

15. 16.

17. 18.

19. 20.

21. 22.

23. 24.

25. Rate of Change The rate of change of the functionis given by Show that

the rate of change can also be written as 26. Rate of Change The rate of change of the function

is given by Show that the rate of change can also be written as


27.

28.

29.

30.

31.

32.

33.

34.

35.

36.

37.

38.

Solving a Trigonometric Equation In Exercises 39–50,solve the equation.

39. 40.

41. 42.

43. 44.

45. 46.

47. 48.

49. 50.

Solving a Trigonometric Equation In Exercises 51–54,find all solutions of the equation in the interval Use a graphing utility to check your answers.

51. 52.

53. 54.

Functions of Multiple Angles In Exercises 55–58, find allsolutions of the multiple-angle equation in the interval

55.

56.57.

58. 3 csc2 5x � �4

cos 4x�cos x � 1� � 0�3 tan 3x � 0

2 sin 2x � �2

[0, 2�.

sin2 x � 2 cos x � 2cos2 x � sin x � 1

2 sin2 x � 3 sin x � �12 cos2 x � cos x � 1

[0, 2�.

csc x � 2 cot x � 0sin x � tan x � 0

sin x�sin x � 1� � 04 cos2 x � 3 � 0

4 tan2 x � 1 � tan2 x3 csc2 x � 4

12 sec x � 1 � 03�3 tan x � 3

4 cos x � 1 � 2 cos xsin x � �3 � sin x

tan x � 1 � 02 sin x � 1 � 0

5.3

tan��

2� x sec x � csc x

csc2��

2� x � 1 � tan2 x

1 � sec��x�sin��x� � tan��x� � �csc x

csc��x�sec��x� � �cot x

�1 � cos x ��sin x�

�1 � cos x

�1 � sin �

1 � sin ��

1 � sin �

�cos ��

cos3 x sin2 x � �sin2 x � sin4 x� cos x

sin5 x cos2 x � �cos2 x � 2 cos 4 x � cos6 x� sin x

cot2 x � cos2 x � cot2 x cos2 x

sin3 � � sin � cos2 � � sin �

sec2 x cot x � cot x � tan x

cos x�tan2 x � 1� � sec x

5.2

�1 � cos x��sin2 x.

cot x.csc2 x � csc xf�x� � csc x � cot x

cot x�sin x.sin�1�2 x cos x.f �x� � 2�sin x

sin3 � � cos3 �

sin � � cos �


sin2 � � sin � cos �

sin��x� cot x

sin��

2� x

tan��

2� x sec x

csc2 x�1 � cos2 x�tan2 ��csc2 � � 1�

sec2��csc2 �


sin �

sec2 x � 1

sec x � 1

1

tan2 x � 1

csc��

2� � � 3, sin � �

2�23

sin��

2� x �

1�2

, sin x � �1�2

tan � �23

, sec � ��13

3

sin x �45

, cos x �35

tan��x�sec��x�

cot��

2� xcsc��

2� x

�1 � tan2 x�1 � cos2 x

1tan x

1sec x

1sin x

1cos x

5.1

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Review Exercises 397

Functions of Multiple Angles In Exercises 59–62, solvethe multiple-angle equation.

59. 60.

61. 62.

Using Inverse Functions In Exercises 63–66, use inversefunctions where necessary to find all solutions of theequation in the interval

63. 64.

65.

66.

Evaluating Trigonometric Functions In Exercises 67–70,find the exact values of the sine, cosine, and tangent ofthe angle.

67. 68.

69. 70.

Rewriting a Trigonometric Expression In Exercises71–74, write the expression as the sine, cosine, or tangentof an angle.

71.

72.

73. 74.

Evaluating a Trigonometric Expression In Exercises75– 80, find the exact value of the trigonometric functiongiven that and (Both and are inQuadrant II.)

75. 76.

77. 78.

79. 80.


81. 82.

83. 84.

85.

86.

Solving a Trigonometric Equation In Exercises 87 and 88,find the solutions of the equation in the interval

87.

88.

Evaluating Functions Involving Double Angles InExercises 89–92, find the exact values of and using the double-angle formulas.

89.

90.

91.

92.

Verifying Trigonometric Identities In Exercises 93–96,use double-angle formulas to verify the identity algebraically. Use a graphing utility to check your resultgraphically.

93.

94.

95.

96.

97. Projectile Motion A baseball leaves the hand of thefirst baseman at an angle of with the horizontal andwith an initial velocity of feet per second. Theball is caught by the second baseman 100 feet away.Find where the range of a projectile is given by

98. Projectile Motion Use the equation in Exercise 97 tofind when a golf ball is hit with an initial velocity of

feet per second and lands 77 feet away.

Reducing a Power In Exercises 99–104, use the power-reducing formulas to rewrite the expression in terms ofthe first power of the cosine.

99. 100.

101. 102.

103. 104.

Using a Half-Angle Formula In Exercises 105–108, usethe half-angle formulas to determine the exact values ofthe sine, cosine, and tangent of the angle.

105. 106.

107. 108.11�

127�

8

112 3015

sin2 2x tan2 2xtan2 4x

sin4 2xcos4 2x

cos4 x sin4 xsin6 x

v0 � 50�

r �132 v0

2 sin 2�.r�

v0 � 80�

sin 4x � 8 cos3 x sin x � 4 cos x sin x

1 � 4 sin2 x cos2 x � cos2 2x

4 sin x cos x � 2 � 2 sin 2x � 2

6 sin x cos x � 3sin 2x

cos u � �2�5

, �

2< u < �

tan u � �29

, �

2< u < �

cos u �45

, 3�

2< u < 2�

sin u �57

, 0 < u <�

2

tan 2ucos 2u,sin 2u,

5.5

cos�x ��

4 � cos�x ��

4 � 1

sin�x ��

2 � sin�x ��

2 � �2

[0, 2�.

sin�� cos � cos �

� tan � � tan �

cos 3x � 4 cos3 x � 3 cos x

sin�� x� � sin xcot��

2� x � tan x

sin�x �3�

2 � cos xcos�x ��

2 � �sin x

cos�u � v�cos�u � v�sin�u � v�tan�u � v�tan�u � v�sin�u � v�

vucos v � � 725.sin u � 4

5

tan 63 � tan 112

1 � tan 63 tan 112

tan 25 � tan 50

1 � tan 25 tan 50

cos 45 cos 120 � sin 45 sin 120

sin 130 cos 50 � cos 130 sin 50

23�

12�

7�

6�

3�

431�

12�

11�

6�

3�

4

345 � 300 � 45285 � 315 � 30

5.4

sec2 x � 6 tan x � 4 � 0

tan2 x � tan x � 12 � 0

3 cos2 x � 5 cos x � 0sin2 x � 2 sin x � 0

[0, 2�.

4 cos2 2x � 3 � 02 sin2 3x � 1 � 0

2 cos 4x � �3 � 02 sin 2x � 1 � 0

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Using a Half-Angle Formula In Exercises 109–112, findthe exact values of and usingthe half-angle formulas.

109.

110.

111.

112.

Using a Half-Angle Formula In Exercises 113–116, usethe half-angle formulas to simplify the expression.

113. 114.

115. 116.

Agriculture In Exercises 117 and 118, a trough forfeeding cattle is 4 meters long and its cross sections areisosceles triangles with two equal sides of meter (see figure). The angle between the equal sides is

117. Write the trough’s volume as a function of

118. Write the volume of the trough as a function of and determine the value of such that the volume ismaximum.

Writing Products as Sums In Exercises 119–122, use theproduct-to-sum formulas to write the product as a sumor difference.

119. 120.

121. 122.

Writing Sums as Products In Exercises 123–126, use thesum-to-product formulas to write the sum or differenceas a product.

123. 124.

125.

126.

Harmonic Motion In Exercises 127–130, a weight isattached to a spring suspended vertically from a ceiling.When a driving force is applied to the system, the weightmoves vertically from its equilibrium position. This motionis described by the model

where is the distance from equilibrium in feet and isthe time in seconds.

127. Write the model in the form

128. Use a graphing utility to graph the model.

129. Find the amplitude of the oscillations of the weight.

130. Find the frequency of the oscillations of the weight.


131. If then

132.

133.

134.

135. Think About It List the reciprocal identities, quotientidentities, and Pythagorean identities from memory.

136. Think About It Is an identity?Explain.

137. Think About It Is any trigonometric equation with aninfinite number of solutions an identity? Explain.

138. Think About It Does the equation have a solution when Explain.

Think About It In Exercises 139 and 140, use the graphsof and to determine how to change to a new function such that

139. 140.

−4

5

�−�

y1

y2

−1

5

�−�

y1y2

y2 � �2 sin x�2y2 � cot2 x

y1 �cos 3x

cos xy1 � sec2��

2� x

y1 � y3.y3

y2y2y1

�a� < �b�?a sin x � b � 0

cos � � �1 � sin2 �

4 sin 45 cos 15 � 1 � �3

4 sin��x� cos��x� � �2 sin 2x

sin�x � y� � sin x � sin y

cos �

2< 0.

�

2< � < �,

y � �a2 � b2 sin�Bt � C�.

ty

y � 1.5 sin 8t � 0.5 cos 8t

cos�x ��

6 � cos�x ��

6

sin�x ��

4 � sin�x ��

4sin 3� � sin 2�cos 5� � cos 4�

cos 6� sin 8�sin 5� sin 4�

4 sin 15 sin 456 sin �

4 cos

�

4

��

��2.

m

12 m

1 2

4 m

θ

�.

12

1 � cos 12xsin 12x

sin 10x1 � cos 10x

�1 � cos 6x2

��1 � cos 8x2

sec u � �6, �

2< u < �

cos u � �27

, �

2< u < �

tan u �43

, � < u <3�

2

sin u �35

, 0 < u <�

2

tan�u/2cos�u/2,sin�u/2,

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Chapter Test 399

5 Chapter Test See www.CalcChat.com for worked-out solutions to odd-numbered exercises.For instructions on how to use a graphing utility, see Appendix A.

Take this test as you would take a test in class. After you are finished, check yourwork against the answers in the back of the book.

1. Given and use the fundamental identities to evaluate the otherfive trigonometric functions of

2. Use the fundamental identities to simplify

3. Factor and simplify

4. Add and simplify

5. Determine the values of for which is true.

6. Use a graphing utility to graph the functions andMake a conjecture about and Verify your result algebraically.

In Exercises 7–12, verify the identity.

7.

8.

9.

10.

11. is an integer.

12.

13. Find the exact value of

14. Rewrite in terms of the first power of the cosine.

15. Use a half-angle formula to simplify the expression

16. Write as a sum or difference.

17. Write as a product.

In Exercises 18–21, find all solutions of the equation in the interval

18.

19.

20.

21.

22. Use a graphing utility to approximate the solutions of the equation accurate to three decimal places.

23. Use the figure to find the exact values of and

24. The index of refraction of a transparent material is the ratio of the speed of lightin a vacuum to the speed of light in the material. For the triangular glass prism inthe figure, and Find the angle for the glass prism given that

n �

sin��

2�

�

2sin

�

2

.

�� 60.n � 1.5

n

tan 2u.cos 2u,sin 2u,

3 cos x � x � 0

csc2 x � csc x � 2 � 0

4 cos2 x � 3 � 0

sin 2� � cos � � 0

tan2 x � tan x � 0

[0, 2�.

sin 3� � sin 4�

4 cos 2� sin 4�

sin 4�

1 � cos 4�.

sin4 x tan2 x

tan 255.

�sin x � cos x�2 � 1 � sin 2x

nsin�n� � �� 1�n sin �,

cos�x ��

2 � �sin x

csc � � sec �

sin � � cos �� cot � � tan �

sec2 x tan2 x � sec2 x � sec4 x

sin � sec � � tan �

y2.y1y2 � csc x.y1 � sin x � cos x cot x

tan � � ��sec2 � � 10 � � < 2�,�,

cos �

sin ��

sin �

cos �.

sec4 x � tan4 x

sec2 x � tan2 x.

csc2 ��1 � cos2 ��.�.

cos � < 0,tan � �32

θ

Prism

Ai r α

Light

Figure for 24

(1, 2)

u x

y

Figure for 23

1111572836_050R.qxd 9/29/10 1:32 PM Page 399



Proofs in Mathematics

Proof

You can use the figures at the right for the proofs of the formulas for In thetop figure, let be the point and then use and to locate the points

and on the unit circle. So, for 2, and 3. For convenience, assume that In the bottom figure, note that arcs and

have the same length. So, line segments and are also equal in length, whichimplies that

Finally, by substituting the values and you obtain The formula for

can be established by considering and using the formulajust derived to obtain

You can use the sum and difference formulas for sine and cosine to prove the formulasfor

Quotient identity

Sum and difference formulas

�

sin u cos v ± cos u sin vcos u cos v

cos u cos v � sin u sin vcos u cos v

�sin u cos v ± cos u sin vcos u cos v � sin u sin v

tan�u ± v� �sin�u ± v�cos�u ± v�

tan�u ± v�.

� cos u cos v � sin u sin v.

cos�u � v� � cos�u � ��v�� cos u cos��v� � sin u sin��v�

u � v � u � ��v�cos�u � v�cos�u � v� � cos u cos v � sin u sin v.y1 � sin v,

y3 � sin u,x1 � cos v,x3 � cos u,x2 � cos�u � v�,

x2 � x3x1 � y3y1.

1 � 1 � 2x2 � 1 � 1 � 2x1x3 � 2y1y3

�x22 � y2

2� � 1 � 2x2 � �x32 � y3

2� � �x12 � y1

2� � 2x1x3 � 2y1y3

x22 � 2x2 � 1 � y2

2 � x32 � 2x1x3 � x1

2 � y32 � 2y1y3 � y1

2

��x2 � 1�2 � �y2 � 0�2 � ��x3 � x1�2 � � y3 � y1�2

BDACBDAC0 < v < u < 2�.

i � 1,xi2 � yi

2 � 1D�x3, y3�C�x2, y2�,B�x1, y1�,vu�1, 0�A

cos�u ± v�.

Sum and Difference Formulas (p. 377)






1 � tan u tan v


1 � tan u tan v

Divide numerator and denominatorby cos u cos v.

x A(1, 0)

u

v u − v

B(x1, y1)C(x2, y2)

D(x3, y3)

y

x

y

A(1, 0)

B(x1, y1)C(x2, y2)

D(x3, y3)

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Proofs in Mathematics 401


Product of fractions

Quotient identity

ProofTo prove all three formulas, let in the corresponding sum formulas.

ProofTo prove the first formula, solve for in the double-angle formula

as follows.

Write double-angle formula.

Divide each side by 2. sin2 u �1 � cos 2u

2

2 sin2 u � 1 � cos 2u

cos 2u � 1 � 2 sin2 u

cos 2u � 1 � 2 sin2 u,sin2 u

�2 tan u

1 � tan2 u�

tan u � tan u1 � tan u tan u

tan 2u � tan�u � u�

� cos2 u � sin2 u� cos u cos u � sin u sin u cos 2u � cos�u � u�

� 2 sin u cos u� sin u cos u � cos u sin usin 2u � sin�u � u�

v � u

�tan u ± tan v

1 � tan u tan v

�

sin ucos u

±sin vcos v

1 � sin ucos u

sin vcos v

�

sin u cos vcos u cos v

±cos u sin vcos u cos v

cos u cos vcos u cos v

� sin u sin vcos u cos v

Double-Angle Formulas (p. 384)

tan 2u �2 tan u

1 � tan2 u

sin 2u � 2 sin u cos u

� 1 � 2 sin2 u

� 2 cos2 u � 1

cos 2u � cos2 u � sin2 u

Power-Reducing Formulas (p. 386)


2cos2 u �

1 � cos 2u2

tan2 u �1 � cos 2u1 � cos 2u

Subtract from andadd to each side.2 sin2 x

cos 2u

Trigonometryand Astronomy

Trigonometry was used byearly astronomers to calculatemeasurements in the universe.Trigonometry was used tocalculate the circumferenceof Earth and the distance fromEarth to the moon. Anothermajor accomplishment inastronomy using trigonometrywas computing distances to stars.

1111572836_050R.qxd 9/29/10 1:32 PM Page 401



In a similar way, you can prove the second formula by solving for in the double-angle formula

To prove the third formula, use a quotient identity, as follows.

ProofTo prove the first formula, let and Then substitute

and in the product-to-sum formula.

The other sum-to-product formulas can be proved in a similar manner.

2 sin�x � y2 cos�x � y

2 � sin x � sin y

sin�x � y2 cos�x � y

2 �12

�sin x � sin y�

sin u cos v �12

�sin�u � v� � sin�u � v��

v � �x � y��2u � �x � y��2y � u � v.x � u � v

�1 � cos 2u1 � cos 2u

�

1 � cos 2u2

1 � cos 2u2

tan2 u �sin2 ucos2 u

cos 2u � 2 cos2 u � 1.

cos2 u

Sum-to-Product Formulas (p. 388)

sin u � sin v � 2 sin�u � v2 cos�u � v

2 sin u � sin v � 2 cos�u � v

2 sin�u � v2

cos u � cos v � 2 cos�u � v2 cos�u � v

2 cos u � cos v � �2 sin�u � v

2 sin�u � v2

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5 analytic trigonometry - verona public schools€¦ · 350 chapter 5 analytic trigonometry...

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