8.3: Algebraic Solutions of Trigonometric Equations

© 2008 Roy L. Gover(www.mrgover.com)

Learning Goals:

•Solve Trigonometric equations using Algebraic methods.

Try ThisUsing your knowledge of Algebra and factoring, solve the equation:

2 0x x

0, 1x x

Check your answer.

ExampleSolve the equation and check the solution:

2sin sin 0x x

Important Idea

Always check your solution(s) in the original equation.

Try ThisUsing your knowledge of Algebra, solve the equation:

2 3 9x

3x

Check your answer.

ExampleSolve the equation and check the solution:

2cos 1 3

Important IdeaSimilar methods are used to solve algebraic and trigonometric equations.

ExampleSolve for all values:

…there are an infinite number of solutions.

2cos 3x

DefinitionPrincipal Values in Degrees:

& tan x: 90 90x

cos x: 0 180x

Principal values insure unique solutions

sin x

DefinitionPrincipal Values in Radians:

sin x & tan x: 2 2x

cos x: 0 x

Principal values insure unique solutions

Try ThisSolve for the principal value:

2cos 1 0x

60x or3

Try ThisSolve for the principal value in degrees:

30x

What is the answer in rads? 6

x

2sin 1 0x

Try ThisSolve for the principal value in radians:

3sin 1x

.340x

Important Idea

When you cannot get exact values for solutions, you must use the inverse trig functions on your calculator.

1sin x

1cos x

1tan x

•

•

•

Try ThisSolve for the principal value in degrees:

8cos 5 0

51.318x

ExampleSolve for all values in degrees:

8cos 5 0

Try This

Solve: 2sin 5 0x

???????

Important Idea

The value of sin x and cos x can never be greater than 1 or less than -1

Range:sin x : [-1,1]

cos x : [-1,1]

Try ThisUsing your knowledge of Algebra and factoring, solve the equation:

2 2 0x x

1, 2x x

Check your answer.

ExampleSolve the equation and check the solution:

2sin sin 2 0x x

Try ThisSolve the equation and check the solution:

2cos cos 2 0x x 0x

Are there other solutions?

Example

Find all solutions of 23sin sin 2 0x x

Try This Solve for the principal values in radians:

22sin sin 1 0x x

6x

or

2

Why is not an answer?

3

2

ExampleFind all solutions in radians:

2tan cos tanx x x

Try ThisFind the principal value solutions in degrees:

2cos sin cosx x x

90x 90x or

ExampleSolve:

210cos 3sin 9 0x x

ExampleSolve:

2sec 5tan 2x x

Important Idea

2 2sin cos 1x x 2 2sin 1 cosx x 2 2cos 1 sinx x

Trigonometric Identities:

2 2tan 1 secx x 2 21 cot cscx x

See section 6.5 of your text for additional trig identities.

Important Idea

2 4

2

b b acx

a

2 0ax bx c for

The quadratic formula:

Try ThisSolve for principal values in radians:

2sec 2 tan 3x x

x=1.220 and x= -.632

Important IdeaWhen solving trig

identities, try one or more of the following:•Factor if possible•Use quadratic formula if you can’t factor•Use identities to write the equation in terms of only one trig function.

Lesson Close

Trignometric equations are used in science and engineering.