1. Ch. 5 – Analytic Trigonometry
5.3 – Solving Trig Equations

2. Solving Trig Equations
When solving trig equations, get the trig function by itself first, then think unit circle or use a calculator to finish!
Ex: Find all solutions of 2sinx – 1 = 0 over the interval [0, 2π).
Solve for sin!
2sinx = 1
sinx = ½
Now we can use inverse trig functions (think unit circle!)
Since sinx = ½ at both and , those are our answers!
Remember: x is an angle here!

3. Solving Trig Equations
Ex: Find all solutions of cosx + 1 = -cosx .
Solve for cos!
1 = -2cosx
- ½ = cosx
Find cos-1 (- ½) to get…
We also want the other solution where cos(x) = -½ , so think unit circle to get …
However, we want ALL SOLUTIONS of x, not just on [0, 2π)!
Since the period of sine and cosine is 2π, if we add/subtract multiples of 2π, we also get solutions for x!
To indicate this in our answer, we add n times the period
n signifies some integer value
Final answer: and

4. Ex: Find all solutions of 3tan2x – 1 = 0 .
Solve for tan!
3tan2x = 1
tan2x = 1/3
tanx =
Now we can use inverse trig functions!
Find the tan-1 of both answers to get …
However, we want ALL SOLUTIONS of x, not just on [0, 2π)!
Since the period of tangent is π, if we add/subtract multiples of π, we also get solutions for x, so…
Final answer:

5. Find all solutions to over the interval [0, 2π).

6. Find all solutions to over the interval [0, 2π).

7. Find all solutions to

8. Find all solutions to

9. Ex: Find all solutions of sinx – sinx cos2x = 1/8 on the interval [0, 2π).
We can use trig identities to simplify the left side of the equation to one trig function!

10. Ex: Find all solutions of 2sin2x – sinx – 1 = 0 on the interval [0, 2π).
Factor first…
(2sinx + 1)(sinx – 1) = 0
Now we have two equations to solve…
Think unit circle to find all values on [0, 2π) that satisfy one of those equations…
All three of these are possible answers!

11. Ex: Find all solutions of 4cos2(3x) – 1 = 0 .
Solve for cos…
4cos2(3x) = 1
cos2(3x) = 1/4
cos(3x) = ± ½
Now we have two equations to solve. However, we won’t divide by 3 until the trig function has left the equation…
Final answers:

12. Ex: Find all solutions of csc (4x) – 2 = 0 on the interval [0, π).
Solve for csc, then flip it to solve for sin…
csc(4x) = 2
sin(4x) = ½
Using unit circle knowledge, we know…
Now find all solutions to these equations on the interval [0, π)…

13. Find all solutions to

14. Find all solutions to over the interval [0, 2π).