5.3 solving trigonometric equations objective: in this lesson you will be learning how to solve...

5.3 Solving Trigonometric Equations

Objective:

In this lesson you will be learning how to solve trigonometric equations

To solve a trigonometric equation, your goal is to isolate the trigonometric function involved in the equation, then find all values of the variable for which the equation is true.

Turn in your Worksheet on Simplifying Trig Expressions if you haven’t yet.

5.3 Solving Trigonometric Equations

Find all of the solutions of in the intervalxx sin2sin

xx sin2sin

2sinsin xx

2sin2 x

2

2sin x

)2,0[

The solutions in the interval  [0,2 )  are5 7

4 

4x and

5.3 Solving Trigonometric Equations

01sin2 x Solve:

1sin2 x

2

1sin x

Original equation

Add 1 to each side

Divide each side by 2

1 1sin

2x 5

, 0,26 6

x on

But, you were asked to find all solutions, not just those on 0,2

5.3 Solving Trigonometric Equations

has the solutions and in the interval .

Remember that since has a period of , there are

infinitely many other solutions that can be written as:

and

2

1sin x

6

x

6

5x )2,0[

siny x 2

nx 2

6 nx

26

5 General solution

5.3 Solving Trigonometric Equations

The equation has infinitely many solutions. Any

angles that are coterminal with are also solutions

to the equation.

2

1sin x

6

5

6

or

sin 1

2

5.3 Solving Trigonometric EquationsSolve:

1tan3 2 x

3

1tan 2 x

1 3tan

3 3x

01tan3 2 x Add 1 to each side

Divide each side by 3

Take the square root of both sides

5

6x n

6

x n

5.3 Solving Trigonometric EquationsYou Try:

Solve: 2tan 3 0x

2,

3 3x n n

5.3 Solving Trigonometric Equations

xxx cot2coscot 2

0cot2coscot 2 xxx

0)2(coscot 2 xx

0cot x 02cos2 x2cos2 x

2cos x

Set each factor equal to 0

2x n

Find a general solution:

Find all solutions in the interval :

2 1 02sin sin

[ , )0 2

5.3 Solving Trigonometric Equations You Try:

7 11, ,

6 6 2x

5.3 Solving Trigonometric Equations

Solve: 2 sin2x + 3 cos x – 3 = 0

52 , 2 , 2

3 3x n n n

5.3 Solving Trigonometric EquationsYou Try:

Solve: 2 sin2x = 2 + cosx on the interval [0,2π).

3 2 4, , ,

2 2 3 3x

Homework

Day 1: 5.3 pg. 364, 7-29 odd