# Solving Trigonometric Equations

How to solve trigonometric equations problems: examples and their solutions.

## Example 1

To factor the left side of the equation,

change the given equation

by using the identities you've learned.

Change cos^{2} *x* into 1 - sin^{2} *x*.

Pythagorean identities

Think 'sin *x*' as a variable

and solve the equation by factoring.

Then 'sin *x* - 1 = 0' or 'sin *x* + 2 = 0'.

Solving a quadratic equation by factoring

Case 1) sin *x* - 1 = 0

sin *x* = 1

So draw *y* = sin *x* (0 ≦ *x* ≦ 2*π*)

and *y* = 1

to find the intersecting point.*x* = *π*/2 is the intersecting point.

So *x* = *π*/2 is the answer of case 1.

Graphing sine functions

Case 2) sin *x* + 2 = 0

sin *x* = -2

But -1 ≦ sin *x* ≦ 1.

So there's no solution in case 2.

So *x* = *π*/2 is the answer.

## Example 2

Move sin *x* to the left side.

Change sin 2*x* into 2 sin *x* cos *x*.

sin 2*A* (double-angle formula)

sin *x* is the GCF.

So sin *x*(2 cos *x* - 1) = 0.

So 'sin *x* = 0' or '2 cos *x* - 1 = 0'

Solving a quadratic equation by factoring

Case 1) sin *x* = 0

So draw *y* = sin *x* (0 ≦ *x* ≦ 2*π*)

and *y* = 0

to find the intersecting points.*x* = 0, *π*, 2*π* are the intersecting points.

So *x* = 0, *π*, 2*π* are the answers of case 1.

Graphing sine functions

Case 2) 2 cos *x* - 1 = 0

cos *x* = 1/2

So draw *y* = cos *x* (0 ≦ *x* ≦ 2*π*)

and *y* = 1/2

to find the intersecting points.

(Think of a right triangle whose cosine is 1/2:

30-60-90 triangle → cos *π*/3 = 1/2)

So *x* = *π*/3, 2*π* - *π*/3 are the intersecting points.

So *x* = *π*/3, 5*π*/3 are the answers of case 2.

Graphing cosine functions

So *x* = 0, *π*/3, *π*, 5*π*/3, 2*π* are the answers.

## Example 3

Change cos 2*x* into 2 cos^{2} *x* - 1.

cos 2*A* (double-angle formula)

Think 'cos *x*' as a variable

and solve the equation by factoring.

Then '2 cos *x* - 1 = 0' or 'cos *x* - 2 = 0'.

Solving a quadratic equation by factoring

Case 1) 2 cos *x* - 1 = 0

cos *x* = 1/2

So draw *y* = cos *x* (0 ≦ *x* ≦ 2*π*)

and *y* = 1/2

to find the intersecting points.

(Think of a right triangle whose cosine is 1/2:

30-60-90 triangle → cos *π*/3 = 1/2)

So *x* = *π*/3, 2*π* - *π*/3 are the intersecting points.

So *x* = *π*/3, 5*π*/3 are the answers of case 1.

Graphing cosine functions

Case 2) cos *x* - 2 = 0

cos *x* = 2

But -1 ≦ cos *x* ≦ 1.

So there's no solution in case 2.

So *x* = *π*/3 and 5*π*/3 are the answers.