5.3 solving trig equations

5.3 SOLVING TRIG EQUATIONS

SOLVING TRIG EQUATIONS Solve the following equation for x:

Sin x = ½

SOLVING TRIG EQUATIONS In this section, we will be solving various

types of trig equations

You will need to use all the procedures learned last year in Algebra II

All of your answers should be angles.

Note the difference between finding all solutions and finding all solutions in the domain [0, 2π)

SOLVING TRIG EQUATIONS Guidelines to solving trig equations:

1) Isolate the trig function

2) Find the reference angle

3) Put the reference angle in the proper quadrant(s)

4) Create a formula for all possible answers (if necessary)

SOLVING TRIG EQUATIONS

1- 2 Cos x = 0

1) Isolate the trig function1- 2 Cos x = 0 + 2 Cos x = + 2 Cos x

1= 2 Cos x 2 2

Cos x = ½


Cos x = ½

2) Find the reference anglex =

3

3) Put the reference angle in the proper quadrant(s) I = 3

IV = 3

5


Cos x = ½

4) Create a formula if necessary

x = 3

x = 35

2n

2n

SOLVING TRIG EQUATIONS Find all solutions to the following equation:

Sin x + 1 = - Sin x+ Sin x + Sin x→ 2 Sin x + 1 = 0

- 1 - 1

→ 2 Sin x = -1→ Sin x = - ½


Sin x = - ½

Ref. Angle: Quad.:6

IV III,

III: 67

Iv: 611

2n

2n

SOLVING TRIG EQUATIONS Find the solutions in the interval [0, 2π) for

the following equation:

Tan²x – 3 = 0

Tan²x = 3Tan x = 3


Tan x = 3

Ref. Angle: Quad.:3

IV III, II, I,

I: 3

II:3

2 III: 34

IV: 35

x = ,3 ,

32 ,

34

35

SOLVING TRIG EQUATIONSSolve the following equations for all real

values of x.

a) Sin x + = - Sin x

b) 3Tan² x – 1 = 0

c) Cot x Cos² x = 2 Cot x

2

SOLVING TRIG EQUATIONS Find all solutions to the following equation:

Sin x + = - Sin x

2 Sin x = -

Sin x = -

2

2

22

x = 45

x = 47

2n

2n


3Tan² x – 1 = 0

Tan² x = 31

Tan x = 3

1

x = 6

65

x =

67

x =

611

x =

2n

2n

2n

2n


Cot x Cos² x = 2 Cot x

Cot x Cos² x – 2 Cot x = 0 Cot x (Cos² x – 2) = 0 Cot x = 0 Cos² x – 2 = 0

Cos² x – 2 = 0 Cos x = 2

Cos x = 0 x = 2

x = 23

2n

2n No Solution


2

SOLVING TRIG EQUATIONSFind all solutions to the following equation.

4 Tan²x – 4 = 0 Tan²x = 1 Tan x = ±1 Ref. Angle =

4

x = 4

43

x =

n

n

SOLVING TRIG EQUATIONS Equations of the Quadratic Type

Many trig equations are of the quadratic type:

2Sin²x – Sin x – 1 = 0 2Cos²x + 3Sin x – 3 = 0

To solve such equations, factor the quadratic or, if that is not possible, use the quadratic formula

SOLVING TRIG EQUATIONSSolve the following on the interval [0, 2π)

2Cos²x + Cos x – 1 = 0

If possible, factor the equation into two binomials.

2x² + x - 1

(2Cos x – 1) (Cos x + 1) = 0Now set each factor equal to zero


2Cos x – 1 = 0 Cos x + 1 = 0Cos x = ½

Ref. Angle: 3

Quad: I, IV

x = ,3

35

Cos x = -1

x =


2Sin²x - Sin x – 1 = 0

(2Sin x + 1) (Sin x - 1) = 0


2Sin x + 1 = 0 Sin x - 1 = 0Sin x = - ½

Ref. Angle: 6

Quad: III, IV

x = ,6

76

11

Sin x = 1

x = 2


2Cos²x + 3Sin x – 3 = 0

Convert all expressions to one trig function

2 (1 – Sin²x) + 3Sin x – 3 = 02 – 2Sin²x + 3Sin x – 3 = 0

0 = 2Sin²x – 3Sin x + 1


2Sin x - 1 = 0 Sin x - 1 = 0Sin x = ½

Ref. Angle: 6

Quad: I, II

x = ,6

65

Sin x = 1

x = 2

0 = 2Sin²x – 3Sin x + 1 0 = (2Sin x – 1) (Sin x – 1)


2Sin²x + 3Cos x – 3 = 0

Convert all expressions to one trig function

2 (1 – Cos²x) + 3Cos x – 3 = 02 – 2Cos²x + 3Cos x – 3 = 0

0 = 2Cos²x – 3Cos x + 1


2Cos x - 1 = 0 Cos x - 1 = 0Cos x = ½

Ref. Angle: 3

Quad: I, IV

x = ,3

35

Cos x = 1

x = 0

0 = 2Cos²x – 3Cos x + 1 0 = (2Cos x – 1) (Cos x – 1)

SOLVING TRIG EQUATIONS The last type of quadratic equation would be

a problem such as:

Sec x + 1 = Tan x

What do these two trig functions have in common?

When you have two trig functions that are related

through a Pythagorean Identity, you can square

both sides.

( )² ²


(Sec x + 1)² = Tan²x Sec²x + 2Sec x + 1 = Sec²x - 1

2 Sec x + 1 = -1 Sec x = -1

Cos x = -1x =

When you have a problem that requires you to square both sides, you must check your answer when you are done!


Sec x + 1 = Tan x x =

1 1 0


Cos x + 1 = Sin x Cos²x + 2Cos x + 1 = 1 – Cos² x

2Cos² x + 2 Cos x = 0Cos x (2 Cos x + 2) = 0

Cos x = 0x = ,

2

(Cos x + 1)² = Sin² x

Cos x = - 1

23

x =


Cos x + 1 = Sin x

x = ,2

,2

3 2Sin 1

2 Cos

1 1 0

23Sin 1

23 Cos

1- 1 0

Sin 1 Cos

0 1 1-

SOLVING TRIG EQUATIONS Equations involving multiply angles

Solve the equation for the angle as your normally would

Then divide by the leading coefficient

SOLVING TRIG EQUATIONS Solve the following trig equation for all

values of x.

2Sin 2x + 1 = 02Sin 2x = -1

Sin 2x = - ½

2x = 2n 6

7 2x = 2n

611

x = n 127

x = n 12

11


0 3 2xTan 3

1-

2xTan

2x

n 4

3

2x

n 4

7

2x

2n 2

3

2x

2n 2

7

Redundant Answer

SOLVING TRIG EQUATIONS Solve the following equations for all values of

x.

a) 2Cos 3x – 1 = 0

b) Cot (x/2) + 1 = 0


2Cos 3x - 1 = 02Cos 3x = 1

Cos 3x = ½

3x = 2n 3

3x = 2n 3

5

x = 32n

9

x = 32n

95


0 1 2xCot

1-

2xCot

2x

n 4

3

2x

2n 2

3

SOLVING TRIG EQUATIONS Topics covered in this section:

Solving basic trig equations Finding solutions in [0, 2π) Find all solutions

Solving quadratic equations Squaring both sides and solving Solving multiple angle equations Using inverse functions to generate answers


Find all solutions to the following equation:

Sec²x – 3Sec x – 10 = 0(Sec x + 2) (Sec x – 5) = 0

Sec x + 2 = 0 Sec x – 5 = 0Sec x = -2Cos x = - ½

x = 2n 3

2

2n 3

4

Sec x = 5Cos x =

51

x =

51Cos 1-

SOLVING TRIG EQUATIONS One of the following equations has solutions

and the other two do not. Which equations do not have solutions.

a) Sin²x – 5Sin x + 6 = 0b) Sin²x – 4Sin x + 6 = 0c) Sin²x – 5Sin x – 6 = 0

Find conditions involving constants b and c that

will guarantee the equation Sin²x + bSin x + c = 0

has at least one solution.

SOLVING TRIG FUNCTIONS Find all solutions of the following equation in

the interval [0, 2π)

Sec²x – 2 Tan x = 41 + Tan²x – 2Tan x – 4 = 0

Tan²x – 2Tan x – 3 = 0(Tan x + 1) (Tan x – 3) = 0

Tan x = -1 Tan x = 3

SOLVING TRIG FUNCTIONS

Tan x = -1 Tan x = 3

47 ,

43 X

x = ArcTan 3

ref. angle: 71.6º

Quad: I, III

x = 71.6º, 251.6º

5.3 solving trig equations

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