P.5Trigonometric Function.

A ray, or half-line, is that portion of a line that starts at a point V on the line and extends indefinitely in one direction. The starting point V of a ray is called its vertex.

V Ray

If two lines are drawn with a common vertex, they form an angle. One of the rays of an angle is called the initial side and the other the terminal side.

Vertex Initial Side

Terminal side

Counterclockwise rotationPositive Angle

Vertex Initial Side

Terminal side

Clockwise rotation Negative Angle

Vertex Initial Side

Terminal side

Counterclockwise rotation Positive Angle

An angle is said to be in

if its vertex is at the origin of a rectangular

coordinate system and its initial side

coincides with with positive - axis.

standard position

x

Initial sideVertex

Terminal side

x

y

Angles are commonly measured in either Degrees or Radians

The angle formed by rotating the initial side exactly once in the counterclockwise direction until it coincides with itself (1 revolution) is said to measure 360 degrees, abbreviated 360.

Initial side

Terminal side

Vertex

One degree is 1

360 revolution., ,1

A is an angle of 90 or 14

revolution.

right angle ,

Initial side

Terminal side

Vertex

90 angle; 14

revolution

A is an angle of 180

or 12

revolution.

straight angle ,

Initial sideTerminal side Vertex

180 angle; 12

revolution

Draw a -135 angle.

x

y

Initial sideVertex

Term

inal

side 135

One minute denoted, is defined as

160

degree.

, ,1

One second denoted, is defined as

second, or 1

3600 degree.

, ,1

160

1 counterclockwise revolution = 360

60 = 1 60 = 1

180°

90°

(a) Straight angle

(12 rotation)

(b) Right angle

(14 rotation)

(c) Acute angle(0° < < 90°)

(d) Obtuse angle(90° < < 180°)

Angles

Consider a circle of radius r. Construct an angle whose vertex is at the center of this circle, called the central angle, and whose rays subtend an arc on the circle whose length is r. The measure of such an angle is 1 radian.

r

r

1 radian

For a circle of radius r, a central angle of radians subtends an arc whose length s is

s r

Find the length of the arc of a circle of radius 4 meters subtended by a central angle of 2 radians.

r 4 meters and =2 radians

s r 4 2 8 meters

1 revolution = 2 radians

180 radians

1 degree =180

radian

1 radian =180

degrees

Convert 135 to radians.

135 135180

radian

3

4 radian

2Convert - radians to degrees.

3

2 2 180

3 3 radians =

120

The unit circle is a circle whose radius is 1 and whose center is at the origin.

Since r = 1:

s r

becomes

s

(0, 1)

(-1, 0)

(0, -1)

(1, 0)

s

y

x

(0, 1)

(-1, 0)

(0, -1)

(1, 0)

t

y

x

P = (a, b)

= sr radians

Also, s = r r

O

r

O r

r s = r

1 radian

= rr = 1 radian

Radian Measure

5-3-47

s

1. To form a reference triangle for , draw a perpendicular from a point P(a, b) on the terminal side of to the horizontal axis.

2. The reference angle is the acute angle (always taken positive) between the terminal side of and the horizontal axis.

a

b

a

b

P(a, b)

(a, b) (0, 0) is always positive

Reference Triangle and Reference Angle

5-4-50

3

1

2

30°

60°

(/6)

(/3)

2

1

1

45°

45°

(/4)

(/4)

30— 60 and 45 Special Triangles

5-4-51

a

b

a

br

a

b

r

a

b

r

a

b

a

b

If is an arbitrary angle in standard position in a rectangular coordinate system and P(a, b) is a point r units from the origin on the terminal side of , then:

sin = br

cos = ar

r = a2 + b2 > 0; P(a, b) is an arbitrary point on the terminal side of , (a, b) (0, 0)

csc = rb , b 0

sec = ra , a 0

cot = ab , b 0

Trigonometric Functions with Angle Domains Alternate Form

5-4-49

tan = , a 0 ba

P(a, b)P(a, b)

P(a, b)

(1, 0) u

v

(a, b)

W(x)sin x = b csc x =

1b b 0

cos x = a sec x = 1a a 0

tan x = ba a

0 cot x = ab b

0

If x is a real number and (a, b) are thecoordinates of the circular point W(x), then:

Circular Functions

5-2-45

(a, b)

W(x)

(1, 0)

a

b

x units arc lengthx rad

Trigonometric CircularFunction Function

sin = sin x

cos = cos x

tan = tan x

csc = csc x

sec = sec x

cot = cot x

If is an angle with radian measure x, then the value of each trigonometricfunction at is given by its value at the real number x.

Trigonometric Functions with Angle Domains

5-4-48

Right Triangle Ratios

Hypotenuse

Opposite

Adjacent

0° < < 90°

HypOppsin

HypAdj

cos

AdjOpp

tan

OppHyp

csc

AdjHyp

sec

OppAdj

cot

5-5-52

Reciprocal Identities

cscsin

1sec

cos

1

cottan

1

Quotient Identities

tansincos

cotcossin

x

y

(a, b)

a < 0, b < 0, r > 0

r

a > 0, b > 0, r > 0a < 0, b > 0, r > 0

a > 0, b < 0, r > 0

I (+, +)

All positive

II ,

sin , csc 0 0

All others negative

III ,

tan , cot 0 0

All others negative

IV ,

cos , sec 0 0

All others negative

x

y

Cast Rule

(C )

(A )

( T )

(S )

Example :Find the value of each of the six trigonometric functions of the angle

.

Adjacent

12 13

c = Hypotenuse = 13

b = Opposite = 12

a b c2 2 2 a2 2 212 13

a2 169 144 25 a 5

a

b

c

Adjacent = 5

Opposite =

Hypotenuse =

12

13

sin OppositeHypotenuse

1213

cos AdjacentHypotenuse

513

tan OppositeAdjacent

125

csc HypotenuseOpposite

1312

sec HypotenuseAdacent

135

cot AdjacentOpposite

512

functions.

ometricsix trigon theof eexact valu theFind

. toscorrespond that circleunit on thepoint the

be 4

15,

4

1let andnumber real a be Let

t

Pt

4

15,41, ba

sin t b 154

cost a 14

tan tba

15

41

415

4

15,41, ba

csc tb

1 115

4

415

4 1515

sec ta

1 11

44

cot tab

1

415

4

115

1515

(0, 1)

(-1, 0)

(0, -1)

(1, 0)

t

y

x

P = (a, b)

Find the exact value of the six trigonometric functions of 45 degrees.

b = 1

c a b2 2 2

a = 1

45

45

c2 2 21 1 2 c 2

c 2

sin sin454

12

22

bc

cos cos454

12

22

ac

tan tan454

11

1 ba

cot cot454

11

1 ab

csc csc454

21

2 cb

sec sec454

21

2 ca

Find the exact value of the six trigonometric functions of 30 and 60 degrees.

60

30

2

30

60

a

b

a

2

2a = 2 so a = 1

c a b2 2 2

2 12 2 2 bb2 2 22 1 4 1 3 b 3

sin sin306

12

ac

cos cos306

32

bc

tan tansin

cos30

630

30

1 23 2

13

33

cot cottan

306

1

30

11 3

3

csc cscsin

306

1

30

11 2

2

sec seccos

306

1

30

13 2

23

2 33

sin sin603

32

bc

cos cos603

12

ac

tan tansin

cos60

360

60

3 21 2

31

3

cot cottan

603

1

60

13

33

csc cscsin

603

1

60

13 2

23

2 33

sec seccos

603

1

60

11 2

2

Use a calculator to find the approximate

value of

(a) sin52 b tan5

(c) ( ) sec

5

and , is so , ofdomain in the

is whenever such that number positive

a is thereif called is function A

pf

p

f

periodic

f p f p

If there is a smallest such number p, this smallest value is called the (fundamental) period of f.

The Graph of y = sin x

0

6

3

2

5 6

3 2

2

0

1 2

3 2

1

1 2

0

1

0

x y

0 2 4 6

1.5

1.5

(0, 0)

6

12,

2 1,

,0

32 1 ,

2 0 ,

Characteristics of the Sine Function

1. The domain is the set of all real numbers.

2. The range consists of all real numbers from -1 to 1, inclusive.3. The sine function is an odd function (symmetric with respect to the origin).

4. The sine function is periodic, with period 2 .5. , ; The - intercepts are ,-2 ,- ,0, ,2

the - intercept is 0.

x

y

6

3 2 2 5 2

2 3 2

7 2

.

, , , , ;

, ,

,

The maximum value is 1 and occurs at

the minimum

value is -1 and occurs at

x

x

Characteristics of the Sine Function

.4

sin2

graph tosin ofgraph theUse

xy

xy

Begin with the basic sine function:

0 2 4 6

1.5

1.5

(0, 0)

6

12,

2 1,

,0

32 1 ,

2 0 ,

0 5

4

4

)sin(xy

0 0,

1,2

0 5

4

4 y x 2sin

2,

2

0 0,

0 5

4

4

0,4

2,

4

3

y 2sin x4

The Graph of y = cos xx y0

6

3

2

2 3

3 2

2

1

3 2

1 2

0

1 2

1

0

1

0 2 4 6

1.5

1.5

y xcos

(0, 1)

2

1,

3

, 1

0,2

3

2 1 ,

Characteristics of the Cosine Function

1. The domain is the set of all real numbers.

2. The range consists of all real numbers from -1 to 1, inclusive.

3. The cosine function is an even function (symmetric with respect to the y-axis).

4. The cosine function is periodic, with

period 2 .5.

, ;

The - intercepts are ,- 3 2 ,- 2 , 2 ,

3 2 the - intercept is 1.

x

y

6

2 0 2 4

3 5

.

, , , , , ;

, ,

,

The maximum value is 1 and occurs at

the minimum

value is -1 and occurs at

x

x

Characteristics of the Cosine Function

0 5

1.5

1.5

2

y xcos

0 5

1.5

1.5

2

y xsin

2cos

xy

The graphs of the sine and cosine functions are called sinusoidal graphs.

If the amplitude and period of

and are given by

0,

sin cosy A x y A x

2

=Period = Amplitude TA

Determine the amplitude and period of

and graph the function.y x 2 2sin ,

y x

y A x

2 2sin

sin

A 2 2,

Amplitude 2 2

T 2 22

0 5

1.5

1.5

2

y xsin

1.57 2.36 6.28

2.5

2.5

1

-1

2

2

2

32

y xsin2

1.57 2.36 6.28

2.5

2.52

-2

2

2

32 2

y x 2 2sin

Find an equation for the graph.

0 2

5

54

-4

-1 1 2

3

Period = T 2 Amplitude 4

T 2

2 22T

y A x sin

Period = T 2 Amplitude A 4

y x 4sin

The Graph of y = tan xx y

3

4

6

0

6

4

3 3

1

58.033

0

58.033

1

73.13

1 0 1

1.73

1.73

3,

3

3

3,

6

(0, 0)

1,4

3,3

tansincos

xxx

?2

as happensWhat

x

?2

as happensWhat

x

5 0 5

3

3x 52 x

2

x 2

x 32

x 52

x 32

Characteristics of the Tangent Function

1. The domain is the set of all real numbers, except odd multiples of 2.

2. The range consists of all real numbers.

3. The tangent function is an odd function (symmetric with respect to the origin).

4. The tangent function is periodic, with

period .5 2. , ; The - intercepts are ,-2 , - , 0, ,

the - intercept is 0.

x

y

6

3 2 2 2 3 2

.

, , , , ,

Vertical asymptotes occurs at

x

Characteristics of the Tangent Function

The graphs of the other three trig functions can be obtained from the graphs of their respective reciprocal functions.

For example:

26csc21

6sin

332

32

6sec23

6cos

5 0 5

3

3

y xsin y x x csc sin1

x 2 x x 0 x x 2

6

12,

6 2,

5 0 5

3

3

y xcos y x x sec cos1

x 3 2 x 2 x 2 x 3 2

3

12,

3 2,

5 0 5

3

3

y x x cot tan1

x 2 x x x 2

6 3,

2 0,

For the graphs of

y A x A x

sin sin

or

y A x A x

cos cos

with > 0,

Amplitude = Period =

Phase shift =

A T 2

Find the amplitude, period, and phase shift of

and graph the function.y x 3 2sin ,

y A x

y x

sin

sin

3 2

A 3 2, ,

Amplitude = A 3 3

Period =2

T 22

Phase shift =

2

0 5

1.5

1.5

y xsin

2

(0,0)

2

3,

xy sin3

(0,0)

12

3,

-1

1

23

y x 3sin

20

,

12

23

, 2sin3 xy

Figure 39: Graphs of the (a) cosine, (b) sine, (c) tangent, (d) secant, (e) cosecant, (f) cotangent functions using and radian measure.

Continued.

Continued.

Figure 45: The general sine curve y = A sin [(2/B)(x – C)]+D, shown for A, B, C, and D positive. (Example 3)

Notes: The domain of the sine function is the set of all real numbers.

1- The domain of the cosine function is the set of all real numbers.

2-The domain of the tangent function is the set of all real numbers except odd multiples of 2 90 .

3-The domain of the secant function is the set of all real numbers except odd multiples of 2 90 .

4-The domain of the cotangent function is the set of all real numbers except integral multiples of

180 .

5- The domain of the cosecant function is the set of all real numbers except integral multiples of 180 .

Let P = (a, b) be the point on the unit circle that corresponds to the angle . Then, -1 < a < 1 and -1 < b < 1.

sin b

1 1sincos a

1 1cos

sin 1 cos 1

Range of the Trigonometric Functions

11

sin

1csc

b

1cscor 1csc

1secor 1sec

11

cos

1sec

a

cot

tan

Periodic Properties

sin sin

cos cos

tan tan

2

2

csc csc

sec sec

cot cot

2

2

4

7cot (b) 390sec (a)

of eexact valu theFind

3

3230sec36030sec390sec (a)

14

3cot

4

3cot

4

7cot )b(

Even-Odd Properties

sin sin

cos cos

tan tan

csc csc

sec sec

cot cot

4cot (b) 30sin (a)

of eexact valu theFind

2

130sin30sin )a(

14

cot4

cot (b)

b a c2 2 2

cb

a

90

b

c

a

c

c

c

2

2

2

2

2

2

122

c

a

c

b

sin cos2 2 1

tan sec2 21

1 2 2 cot csc

Given that cos and is an acute angle,

find the exact value of each of the remaining

five trigonometric function of

14

.

sin cos2 2 1

14

1sin

2

2

2

2

4

11sin

2

4

11sin

sin 11

16154

cos ; sin 14

154

151

4

4

15

41

415

cos

sintan

cscsin

1 115

4

415

4 1515

cottan

1 115

1515

seccos

1 11

44

Sum and Difference Formulas for Cosines

cos cos cos sin sin

cos cos cos sin sin

Find the exact value of cos .105

cos cos105 60 45

cos cos sin sin60 45 60 45

2

2

2

3

2

2

2

1

24

64

2 64

Cofunction Identities

cos2

sin

sin2

cos

Sum and Difference Formulas for Sines

sin sin cos cos sin

sin sin cos cos sin

.12

sin of eexact valu theFind

34sin

12sin

sin cos sin cos 4 3 3 4

2

2

2

3

2

1

2

2

24

64

2 64

cos (c) sin (b) cos (a)

of eexact valu thefind ,2,3

1cos and

;2,2

1sin known that isIt

sin cos2 2 1

cos sin 1 2 1 1 2 2

1 1 4 3 4 32

-1

3

a b r2 2 2

( ) 1 32 2 2b

b2 8

b 2 2

2 2

sin 2 23

sin cos sin cos 12

32

2 23

13

cos cos cos sin sin

3

22

2

1

3

1

2

3

6

22

6

3

3 2 26

Sum and Difference Formulas for Tangents

tantan tan

tan tan

1

tantan tan

tan tan

1

5

3cos

13

12sintan of eexact valu theFind 11

sin ,11213 2 2

sin , 1213 2 2

0 ,

5

3cos 1

0 ,5

3cos

5

1213

tan 125

4

-3

5

tan 43

tan tan 125

43

5

3cos

13

12sintan 11 tan

tan tantan tan

1

125

43

1 125

43

5615

115

5633

Double-Angle Formulas

sin sin cos

cos cos sin

cos sin

cos cos

2 2

2

2 1 2

2 2 1

2 2

2

2

If find the exact value of

(a) sin 2 (b) cos 2

sin , ,

513

2

135

a b r2 2 2 5 132 2 2 b

b2 169 25 144 b 12

-12 cos br

1213

sin cos 513

1213

sin sin cos2 2

13

12

13

52 120

169

cos cos sin2 2 2 22

13

5

13

12

144169

25169

119169

Double-angle Formula for Tangent

tantan

tan2

2

1 2

Variations of the Double Angle Formula’s

cos sin2 1 2 2

2 1 22sin cos

sincos2 1 22

cos cos2 2 12

2 1 22cos cos

coscos2 1 22

Variations of the Double Angle Formula’s

tansin

cos2

2

2

1 22

1 22

cos

cos

1 21 2

coscos

tancoscos

2 1 21 2

Half-Angle Formulas

sincos

coscos

tancoscos

21

2

21

2

211

where the + or sign is determined by

the quadrant of the angle 2

.

If find the exact value of

(a) sin2

(b) cos2

(c) tan2

csc , ,

32

32

2 2

34

so 2

lies in Quadrant II

csc = rb

32

a b r a2 2 2 2 2 22 3 , so

a2 9 4 5 a 5

cos ar

53

(a) sincos

21

2

1 5

32

3 56

(b) coscos

21

2

1 5

32

3 56

(c) tancoscos

2

11

1 53

1 53

3 53 5

cos1

sin

sin

cos1

2tan

Half-Angle Formulas for tan 2

Product-to-Sum Formulas

sin sin cos cos

cos cos cos cos

sin cos sin sin

121212

Express each product as a sum containing

only sines and cosines.

(a) (b) sin sin sin cos3 6 3 6

(a) sin sin cos cos3 612

3 6 3 6

12

3 9cos cos

12

3 9cos cos

(b) sin cos sin sin3 612

3 6 3 6

12

9 3sin sin

12

9 3sin sin

Sum-to-Product Formulas

sin sin sin cos 2

2 2

sin sin sin cos 2

2 2

cos cos cos cos 2

2 2

cos cos sin sin 2

2 2

Express each sum or difference as a product

of sines and / or cosines:

(a) sin4 + sin3 (b) cos5 cos3

(a) sin4 + sin3 = 2sin4 + 3

2

cos4 3

2

272 2

sin cos

(b) cos5 cos3 = 2

5 32

5 32

sin sin

2 4sin sin

cos coscos cos

tan tan4 84 8

2 6

Establish the identity:

cos coscos cos

sin sin

cos cos

4 84 8

24 8

24 8

2

24 8

24 8

2

sin sin

cos cos6 2

6 2

tan tan6 2

tan tan2 6

The End

Good Luck

Reciprocal Identities

csc x = 1

sin x sec x = 1

cos x cot x = 1

tan x

Quotient Identities

tan x = sin xcos x cot x =

cos xsin x

Identities for Negatives

sin(–x) = –sin x cos(–x) = cos x tan(–x) = –tan x

Pythagorean Identities

sin2x + cos2x = 1 tan2x + 1 = sec2x 1 + cot2x = csc2x

Summery

6-1-64

Sum Identities

sin(x + y) = sin x cos y + cos x sin y

cos( x + y) = cos x cos y – sin x sin y

tan(x + y) = tan x + tan y

1 – tan x tan y

Difference Identities

sin(x – y) = sin x cos y – cos x sin y

cos( x – y) = cos x cos y + sin x sin y

tan(x – y) = tan x – tan y

1 + tan x tan y

Cofunction Identities

Replace 2 with 90° if x is in degrees.

cos

2 – x = sin x sin

2 – x = cos x tan

2 – x = cot x

Sum, Difference, and Cofunction Identities

6-2-66

Double-Angle Identities

sin 2x = 2 sin x cos x

cos 2x = cos2x – sin2x = 1 – 2 sin2x = 2 cos2x – 1

tan 2x = 2 tan x

1 – tan2 x =

2 cot xcot2 x – 1

= 2

cot x – tan x

Half-Angle Identities

sinx2 = ±

1 – cos x2

cosx2 = ±

1 + cos x2

tanx2 = ±

1 – cos x1 + cos x =

sin x1 + cos x =

1 – cos xsin x

where the sign is determined by the quadrant in which x2 lies.

Double-Angle and Half-Angle Identities

6-3-67

sin x cos y = 12 [sin (x + y ) + sin (x – y )]

cos x sin y = 12 [sin (x + y ) – sin (x – y )]

sin x sin y = 12 [cos(x – y ) – cos(x + y )]

cos x cos y = 12 [cos(x + y ) + cos(x – y )]

Product-Sum Identities

sin x + sin y = 2 sin x + y

2 cos x – y

2

sin x – sin y = 2 cos x + y

2 sin x – y

2

cos x + cos y = 2 cos x + y

2 cos x – y

2

cos x – cos y = –2 sin x + y

2 sin x – y

2

Sum-Product Identities

Copyright © 2000 by the McGraw-Hill Companies, Inc. 6-4-68