1. Analytic Trigonometry
Chapter 7
Analytic Trigonometry

2. Section 7 Trigonometric Equations

3. Almost always we will specifically be looking for solutions of trig equations for 0 ≤ 𝜃 ≤ 2𝜋 The goal in solving trig equations is to get the trig function and what you are taking that trig function of by itself.

4. Examples 1-4: 1  2sin𝜃 + 3 = 2 2  cos(2𝜃) = ½ 3 tan(𝜃/2 + 𝜋/3) = 1 4  4cos2𝜃 = 1 What is the function and what are you taking the function of? Is it by itself? 1  2  3  4  For the examples above, you need to get the ”function of” part alone FIRST, then solve for 𝜃.

5. Important Notes: Make sure trig function of ( ) is by itself, everything else is on the other side of the equation Then pretend nothing is happening to 𝜃 and think where does cos 𝜃 = ?, sin 𝜃 = ?, tan 𝜃 = ? happen on the unit circle Take what is in ( ) = unit circle value + period x k Solve for 𝜃  use -1, 0, 1, etc. for k and evaluate until you no longer have answers between 0 - 2𝜋

6. Example 1: 2sin𝜃 + 3 = 2, 0 ≤ 𝜃 ≤ 2𝜋

7. k represents the number of complete revolutions 2𝜋 = period = length of when sine and cosine repeat For the previous example, because you are not doing anything directly to 𝜃, you want to be able to change k and stay between 0 < 𝜃 < 2𝜋 so your only answers are: If the question was to solve in general, then you would leave you answer as:

8. Example 2: cos(2𝜃) = ½, 0 ≤ 𝜃 ≤ 2𝜋

10. Example 3: tan(𝜃/2 + 𝜋/3) = 1, 0 ≤ 𝜃 ≤ 2𝜋

12. EXIT SLIP