Warm up• If sin θ= and find the exact value of

each function.

1. cos 2θ

2. sin 2θ

3. Use the half-angle identity to find the exact value of the function: cos

5

2

12

5

20

Lesson 7-5 Solving Trigonometric Equations

Objective: To solve trigonometric equations and inequalities

A trigonometric equation is an equation that contains a trigonometric function. For example:

To solve a trig equation, we use algebra to isolate the trig function on one side of the equal sign.

2 2sin cos 1x x 2sin 1 0x 2tan 1 0x

Example

2sin 1 0x

1sin2 x

2

1sin x

If the variable is not restricted there is an infinite number of solutions.

kk 2

6

5,2

6

If principal values are require then only answers in quadrants I and IV work.

306or

Example• Solve

– Use the Pythagorean Identity

• factor the trinomial

0cos2sin33 2 xxxx 22 sin1cos

0)sin1(2sin33 2 xx

0sin22sin33 2 xx

0sin2sin31 2 xx

01sin3sin2 2 xx

0)1)(sin1sin2( xx

2

1sin

01sin2

x

x

1sin

01sin

x

x

6

5

6

orx

2

x

)2,0[

Example• sin x = cos x dividing each side by cos x

produces a different trig function on the left. This is what you want.

• sin2 x= sin x dividing this does not produce a new function. Because you lose the squared term you will miss one of the answers.

x

x

x

x

cos

cos

cos

sin

1tan x

1sinsin

sin

sin

sin 2

xx

x

x

x0)1(sinsin

0sinsin 2

xx

xx

1sin0sin xx

Practice• Solve for principal values

of x. Express solutions in degree. 01sinsin2 2 xx

Practice• Solve• (Hint: Factor by grouping)

03cos32sin2sincos4 xxxx