Chapter 5.3

Give an algebraic expression that represents the sequence of numbers. Let n be the natural numbers (1, 2, 3, …).

2, 4, 6, …

1, 3, 5, …

7, 10, 13, 16, …

9, 14, 19, 24, …

…45,135,225,315,…

…60,120,240,300,…

5.3 Solving Trigonometric Equations

In this chapter 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.

In other words, get the trigonometric function to one side by itself. Use standard algebra such as collecting like terms and factoring to do this.

5.3 Solving Trigonometric Equations01sin2 xFor Example:1sin2 x

2

1sin x

To solve for x, note that the equation 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[ xsin

2nx

26

nx 2

6

5 General solution

Original equation

Add 1 to each side

Divide each side by 2

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 EquationsCollecting like terms

Find all of the solutions of in the intervalxx sin2sin

)2,0[

xx sin2sin

2sinsin xx

2sin2 x

2

2sin x

The solutions in the interval are

and

)2,0[

4

5x

4

7x

5.3 Solving Trigonometric EquationsTry #17 pg.3641

Find all of the solutions of the equation in the interval

algebraically.

)2,0[

03tan x

5.3 Solving Trigonometric EquationsExtracting Square Roots

Solve:

1tan3 2 x

3

1tan2 x

3

1tan x

01tan3 2 x Add 1 to each side

Divide each side by 3

Take the square root of both sides

Tan x has a period of so first find all of the solution in the

interval [0, ). These are and .Add multiples of

to get the general form and

6

x

6

5x

nx

6

5nx

6

5.3 Solving Trigonometric EquationsTry #19 pg.3642

Find all of the solutions of the equation in the interval

algebraically.

)2,0[

02csc2 x

5.3 Solving Trigonometric EquationsFactoring

Solve: xxx cot2coscot 2 0cot2coscot 2 xxx0)2(coscot 2 xx

0cot x 02cos2 x2cos2 x2cos x

Set each factor equal to 0

The equation cot x=0 has the solution in the interval (0, ). No solution is obtained for because are outside the range of the cosine function. Because cot x has a period of the general form of the solution is obtained by adding multiples of to

get where n is an integer.

2

x

2cos x 2

nx

2

5.3 Solving Trigonometric EquationsTry #21 pg.3643

Find all of the solutions of the equation in the interval

algebraically.

)2,0[ xx tantan3 3

Find all solutions of in the interval 2 1 02sin sin [ , )0 2

Now attempt #28 on p.364

Find all solutions of in the interval cos sin 1 [ , )0 2

Now attempt #26 on p. 364

Solve 2 3 1 0cos x

Now attempt #39 on p.365

Solve 323 0tan

x

Now attempt #45 on p.365