Solving Trigonometric Equations

First Degree Trigonometric Equations:

• These are equations where there is one kind of trig function in the equation and that function is raised to the first power.

1)sin(2 x

Steps for Solving:

• Isolate the Trigonometric function.

• Use exact values to solve and put answers in terms of radians.

• If the answer is not an exact value, then use inverse functions on your calculator to get answers

1)sin(2 x

1sin( )

2x

Now figure out where sin = -1/2 on the unit circle.

1 7 11sin

2 6 6at and

Complete the List of Solutions:

• If you are not restricted to a specific interval and are asked to give the general solutions then remember that adding on any integer multiple of 2π represents a co-terminal angle with the equivalent trigonometric ratio.

k

kx

Solutions

26

11

26

7

:

Where k is an integer and gives all the coterminal angles of the solution.

Practice

• Solve the equation. Find the general solutions

3 csc 2 0

3 csc 2

csc 2

3

22 , 2

3 3k k

3sin

2whichmeans that

Second Degree Trigonometric Equations:

• These are equations that have one kind of Trigonometric function that is squared in the problem.

• We treat these like quadratic equations and attempt to factor or we can use the quadratic formula.

2: 4sin ( ) 1 0 int [0,2 )Solve x over the erval

(2sin 1)(2sin 1) 0x x

This is a difference of squares and can factor

Solve each factor and you should end up with 4 solutions

1 1sin sin

2 2x and x

5 7 11, , ,

6 6 6 6x

Practice

2tan 2 tan 1x x 2tan 2 tan 1 0x x

(tan 1)(tan 1) 0x x

Find the general solutions for

tan 1x 3 7

,4 4

x k k

Writing in terms of 1 trig fnc

• If there is more than one trig function involved in the problem, then use your identities.

• Replace one of the trig functions with an identity so there is only one trig function being used

Solve the following 22cos sin 1 0x x

2

2

2

2

2(1 sin ) sin 1 0

2 2sin sin 1 0

2sin sin 1 0

2sin sin 1 0

x x

x x

x x

x x

Replace cos2 with 1-sin2

(2sin 1)(sin 1) 0

1sin sin 1

27 11

2 , 2 , 26 6 2

x x

x and x

x k x k x k

Solving for Multiple Angles

• Multiple angle problems will now have a coefficient on the x, such as sin2x=1

• Solve the same way as previous problems, but divide answers by the coefficient

• For general solutions divide 2 by the coefficient for sin and cos. Divide by the coefficient for tan and cot.

Find the general solutions for

sin 3 1x 3

32

x

322

3 3

kx

sin 3x +2= 1

2

2 3x k

Practice

2cos 4 3 0Solve x

3cos 4

2x

5 74 4

6 6x and x

5 7,

24 2 24 2

k kx x