1. Solving Trigonometric Equations
Section 5.3
Precalculus PreAP/Dual, Revised ©2017
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§5.3: Solving Trig Equations

2. §5.3: Solving Trig Equations
Steps
Isolate the trig function. [ex. 𝟐𝐬𝐢𝐧 𝒙 =𝟏]
Make sure to apply the inverse restrictions, if necessary. I
Give all answers [with a +𝟐𝝅𝒏 or +𝝅𝒏 on all answers]
+𝝅𝒏 will come from tan or cot
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§5.3: Solving Trig Equations

3. §5.3: Solving Trig Equations
Example 1
Find all solutions and solve without a calculator for 𝐬𝐢𝐧 𝒙= 𝟐 𝟐
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§5.3: Solving Trig Equations

4. §5.3: Solving Trig Equations
Example 2
Find all solutions and solve without a calculator for 𝟑 𝐜𝐬𝐜 𝒙−𝟐=𝟎
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§5.3: Solving Trig Equations

5. §5.3: Solving Trig Equations
Example 3
Find all solutions and solve without a calculator for 𝐬𝐢𝐧 𝒙+ 𝟐 =−𝐬𝐢𝐧 𝒙
2/4/ :16 PM
§5.3: Solving Trig Equations

6. §5.3: Solving Trig Equations
Example 3
Find all solutions and solve without a calculator for 𝐬𝐢𝐧 𝒙+ 𝟐 =−𝐬𝐢𝐧 𝒙
2/4/ :16 PM
§5.3: Solving Trig Equations

7. §5.3: Solving Trig Equations
Your Turn
Find all solutions and solve without a calculator for 𝐜𝐨𝐬 𝒙− 𝟑 =−𝐜𝐨𝐬 𝒙
2/4/ :16 PM
§5.3: Solving Trig Equations

8. §5.3: Solving Trig Equations
Example 4
Find all solutions and solve without a calculator for 𝟑 𝐭𝐚𝐧 𝟐 𝒙−𝟏=𝟎
2/4/ :16 PM
§5.3: Solving Trig Equations

9. §5.3: Solving Trig Equations
Example 4
Find all solutions and solve without a calculator for 𝟑 𝐭𝐚𝐧 𝟐 𝒙−𝟏=𝟎
2/4/ :16 PM
§5.3: Solving Trig Equations

10. §5.3: Solving Trig Equations
Example 5
Find all solutions and solve without a calculator for 𝐜𝐨𝐭 𝒙 𝐜𝐨𝐬 𝟐 𝒙=𝟐 𝐜𝐨𝐭 𝒙
2/4/ :16 PM
§5.3: Solving Trig Equations

11. §5.3: Solving Trig Equations
Example 6
Find all solutions and solve without a calculator for 𝟐𝐜𝐨𝐬 𝟐 𝒙=𝐜𝐨𝐬 𝒙
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§5.3: Solving Trig Equations

12. §5.3: Solving Trig Equations
Your Turn
Find all solutions and solve without a calculator for 𝐬𝐢𝐧 𝟐 𝒙=𝟐 𝐬𝐢𝐧 𝒙
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§5.3: Solving Trig Equations

13. 𝑼-Substitution for Trigonometry
Try to get into one trig function
As a form of substitution, plug the 𝒖 in for the variable with trig functions
Factor the equation with the substitution
Replacing the 𝒖 back to the original trig function and solve
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§5.3: Solving Trig Equations

14. §5.3: Solving Trig Equations
Example 7
Find all solutions and solve without a calculator for 𝟐𝐬𝐢𝐧 𝟐 𝒙+𝟑𝐜𝐨𝐬 𝒙−𝟑=𝟎
2/4/ :16 PM
§5.3: Solving Trig Equations

15. §5.3: Solving Trig Equations
Example 7
Find all solutions and solve without a calculator for 𝟐𝐬𝐢𝐧 𝟐 𝒙+𝟑𝐜𝐨𝐬 𝒙−𝟑=𝟎
2/4/ :16 PM
§5.3: Solving Trig Equations

16. §5.3: Solving Trig Equations
Example 7
Find all solutions and solve without a calculator for 𝟐𝐬𝐢𝐧 𝟐 𝒙+𝟑𝐜𝐨𝐬 𝒙−𝟑=𝟎
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§5.3: Solving Trig Equations

17. §5.3: Solving Trig Equations
Example 8
Find all solutions and solve without a calculator for 𝐜𝐨𝐭 𝟐 𝒙+𝐜𝐬𝐜 𝒙−𝟓=𝟎
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§5.3: Solving Trig Equations

18. §5.3: Solving Trig Equations
Your Turn
Find all solutions and solve without a calculator for 𝟑𝐬𝐞𝐜 𝟐 𝒙−𝟐 𝐭𝐚𝐧 𝟐 𝒙−𝟒=𝟎
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§5.3: Solving Trig Equations

19. §5.3: Solving Trig Equations
Example 9
Find all solutions and solve without a calculator for 𝐬𝐢𝐧 𝟐𝒙=− 𝟏 𝟐
2/4/ :16 PM
§5.3: Solving Trig Equations

20. §5.3: Solving Trig Equations
Example 9
Find all solutions and solve without a calculator for 𝐬𝐢𝐧 𝟐𝒙=− 𝟏 𝟐
2/4/ :16 PM
§5.3: Solving Trig Equations

21. §5.3: Solving Trig Equations
Example 10
Find all solutions and solve without a calculator for 𝟐 𝐜𝐨𝐬 𝟑𝒕−𝟏=𝟎
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§5.3: Solving Trig Equations

22. §5.3: Solving Trig Equations
Example 11
Find all solutions and solve without a calculator for 𝟑 𝐭𝐚𝐧 𝒙 𝟐 +𝟑=𝟎 from 𝟎, 𝝅
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§5.3: Solving Trig Equations

23. §5.3: Solving Trig Equations
Example 11
Find all solutions and solve without a calculator for 𝟑 𝐭𝐚𝐧 𝒙 𝟐 +𝟑=𝟎 from 𝟎, 𝝅
2/4/ :16 PM
§5.3: Solving Trig Equations

24. §5.3: Solving Trig Equations
Your Turn
Find all solutions and solve without a calculator for 𝐜𝐬𝐜 𝟐𝒙− 𝟐 𝟑 =𝟎
2/4/ :16 PM
§5.3: Solving Trig Equations

25. §5.3: Solving Trig Equations
Assignment
Page odd, odd, 31
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§5.3: Solving Trig Equations