1. 7 Analytic Trigonometry
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2. Copyright © Cengage Learning. All rights reserved.
7.3
SOLVING TRIGONOMETRIC EQUATIONS
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3. What You Should Learn
Use standard algebraic techniques to solve trigonometric equations.
Solve trigonometric equations of quadratic type.
Solve trigonometric equations involving multiple angles.
Use inverse trigonometric functions to solve trigonometric equations.

4. Example 1 – Collecting Like Terms
Solve Solution: Begin by rewriting the equation so that sin x is isolated on one side of the equation.
Write original equation.
Add sin x to each side.
Subtract from each side.

5. Example 1 – Solution
cont’d
Because sin x has a period of 2, first find all solutions in the interval [0, 2). These solutions are x = 5 /4 and x = 7 /4. Finally, add multiples of 2 to each of these solutions to get the general form and where n is an integer.
Combine like terms.
Divide each side by 2.
General solution

6. Example 2-3

7. Example 4 – Factoring an Equation of Quadratic Type
Find all solutions of 2 sin2 x – sin x – 1 = 0 in the interval [0, 2). Solution: Begin by treating the equation as a quadratic in sin x and factoring. 2 sin2 x – sin x – 1 = 0 (2 sin x + 1)(sin x – 1) = 0
Write original equation.
Factor.

8. Example 4 – Solution
cont’d
Setting each factor equal to zero, you obtain the following solutions in the interval [0, 2). 2 sin x + 1 = 0 and sin x – 1 = 0

9. Example 5 -6

10. Example 7 – Functions of Multiple Angles
Solve 2 cos 3t – 1 = 0. Solution: 2 cos 3t – 1 = 0 2 cos 3t = 1
Write original equation.
Add 1 to each side.
Divide each side by 2.

11. Example 7 – Solution
cont’d
In the interval [0, 2), you know that 3t =  /3 and 3t = 5 /3 are the only solutions, so, in general, you have and Dividing these results by 3, you obtain the general solution where n is an integer.
General solution

12. Example 9 – Using Inverse Functions
Solve sec2 x – 2 tan x = 4. Solution: sec2 x – 2 tan x = tan2 x – 2 tan x – 4 = 0 tan2 x – 2 tan x – 3 = 0 (tan x – 3)(tan x + 1) = 0
Write original equation.
Pythagorean identity
Combine like terms.
Factor.

13. Example 9 – Solution
cont’d
Setting each factor equal to zero, you obtain two solutions in the interval (– /2,  /2). [Recall that the range of the inverse tangent function is (– /2,  /2).] tan x – 3 = 0 and tan x + 1 = 0 tan x = 3 tan x = –1 x = arctan 3

14. Example 9 – Solution
cont’d
Finally, because tan x has a period of , you obtain the general solution by adding multiples of  x = arctan 3 + n and where n is an integer. You can use a calculator to approximate the value of arctan 3.
General solution

15. Using Sum and Difference Formulas

16. Using Sum and Difference Formulas

17. Example 1 – Evaluating a Trigonometric Function
Find the exact value of . Solution: To find the exact value of , use the fact that Consequently, the formula for sin(u – v) yields

18. Example 1 – Solution
cont’d
Try checking this result on your calculator. You will find that 

19. Using Sum and Difference Formulas
Example 5 shows how to use a difference formula to prove the cofunction identity = sin x.

20. Example 7 – Solving a Trigonometric Equation
Find all solutions of
in the interval . Solution: Using sum and difference formulas, rewrite the equation as

21. Example 7 – Solution So, the only solutions in the interval are and
cont’d
So, the only solutions in the interval are
and