1. 5.3 Solving Trigonometric Equations

2. Simple trigonometric equation
Solve
Solutions:
General Solutions:

3. Techniques to Solve Trigonometric Equations:
Combining like terms
Factoring
Extracting square roots
Factoring quadratic types equations
Rewriting with single variable
Converting to quadratic type
Functions of multiple angles
Using inverse functions

4. Combining Like terms
Example 1:

5. Extracting Square Roots

6. Factoring Solve Can we divide both sides by cot x ? No Solutions
Why cant we divide by cotx?
No Solutions
General Solution

7. Trigonometric Equations of Quadratic type
Quadratic in sin x:
Quadratic in csc x:
To solve quadratic equations, factor or use Quadratic Formula

8. Factoring Quadratic Equation
Set each factor equal to zero

9. Changing to a Single Trigonometric Function
Solve
Use Pythagorean Identity
Factor

10. Changing to a Single Trigonometric Function (continued)
Set each factor equal to zero
Solve for x
General Solution

11. Converting to Quadratic Type
Square both sides
Produces extraneous solutions
Verify possible solutions
Solve
Square both sides
Pythagorean Identity

12. Converting to Quadratic Type (continued)
Combine like terms
Factor
Set factors equal to zero and solve for x

13. Verifying Solutions: cos x+1 = sin x
Check for
Substitute π/2 for x
Solution checks
Solution does not check

14. Verifying Solutions: cos x+1 = sin x (continued)
Check for
Of the three possible solutions x=3π/2 is extraneous
In the interval [0,2 π], there are two solutions:
x = π/2 and x = π
Solution checks

15. The end