1. Double Angle Identities
Section 9.3
Pre-Calculus AB PreAP/Dual, Revised ©2014
5/20/2019 6:17 PM
9.3A: Double Angles Identity

2. Double Angle Identities
5/20/2019 6:17 PM
9.3A: Double Angles Identity

3. 9.3A: Double Angles Identity
Proof
Proof of Sin 2x
5/20/2019 6:17 PM
9.3A: Double Angles Identity

4. 9.3A: Double Angles Identity
Proof
Proof of Cos 2x
5/20/2019 6:17 PM
9.3A: Double Angles Identity

5. 9.3A: Double Angles Identity
Proof
Proof of Cos 2x (Second Part)
5/20/2019 6:17 PM
9.3A: Double Angles Identity

6. 9.3A: Double Angles Identity
Proof
Proof of Cos 2x (Third Part)
5/20/2019 6:17 PM
9.3A: Double Angles Identity

7. 9.3A: Double Angles Identity
Proof
Proof of Tan 2x (Third Part)
5/20/2019 6:17 PM
9.3A: Double Angles Identity

8. Steps for Double Angle Identity
Simplify the equation
Substitute any identities with a double angle identity
Factoring
U-Substitution
It is generally assumed if solving for sin, use the sine identity and same goes with cosine identity
Make the equation equal to zero and solve
5/20/2019 6:17 PM
9.3A: Double Angles Identity

9. 9.3A: Double Angles Identity
Example 1
Use the double angle formula to solve, 𝐬𝐢𝐧 𝟐𝒙−𝐬𝐢𝐧𝒙=𝟎
5/20/2019 6:17 PM
9.3A: Double Angles Identity

10. 9.3A: Double Angles Identity
Example 1
Use the double angle formula to solve, 𝐬𝐢𝐧 𝟐𝒙−𝐬𝐢𝐧𝒙=𝟎
5/20/2019 6:17 PM
9.3A: Double Angles Identity

11. 9.3A: Double Angles Identity
Example 2
Use the double angle formula to solve, 𝐬𝐢𝐧 𝟐𝒙+𝐜𝐨𝐬 𝒙=𝟎
5/20/2019 6:17 PM
9.3A: Double Angles Identity

12. 9.3A: Double Angles Identity
Your Turn
Use the double angle formula to solve, 𝐜𝐨𝐬 𝟐𝒙−𝐬𝐢𝐧𝒙=𝟏
5/20/2019 6:17 PM
9.3A: Double Angles Identity

13. 9.3A: Double Angles Identity
Example 3
Use the double angle formula to solve, 𝒚=𝟒 𝐜𝐨𝐬 𝟐 𝒙−𝟐 over the interval [0, π]
5/20/2019 6:17 PM
9.3A: Double Angles Identity

14. 9.3A: Double Angles Identity
Example 3
Use the double angle formula to solve, 𝒚=𝟒 𝐜𝐨𝐬 𝟐 𝒙−𝟐 over the interval [0, π]
5/20/2019 6:17 PM
9.3A: Double Angles Identity

15. 9.3A: Double Angles Identity
Example 3
Use the double angle formula to solve, 𝒚=𝟒 𝐜𝐨𝐬 𝟐 𝒙−𝟐 over the interval [0, π]
5/20/2019 6:17 PM
9.3A: Double Angles Identity

16. 9.3A: Double Angles Identity
Example 4
Use the double angle formula to solve, 𝐬𝐢𝐧 𝒙=𝐬𝐢𝐧 𝒙 𝟐
5/20/2019 6:17 PM
9.3A: Double Angles Identity

17. 9.3A: Double Angles Identity
Example 4
Use the double angle formula to solve, 𝐬𝐢𝐧 𝒙=𝐬𝐢𝐧 𝒙 𝟐
5/20/2019 6:17 PM
9.3A: Double Angles Identity

18. 9.3A: Double Angles Identity
Example 4
Use the double angle formula to solve, 𝐬𝐢𝐧 𝒙=𝐬𝐢𝐧 𝒙 𝟐
5/20/2019 6:17 PM
9.3A: Double Angles Identity

19. 9.3A: Double Angles Identity
Example 5
Use the double angle formula to solve, 𝐜𝐨𝐬 𝟐𝒙+𝐬𝐢𝐧 𝒙=𝟎
5/20/2019 6:17 PM
9.3A: Double Angles Identity

20. 9.3A: Double Angles Identity
Your Turn
Use the double angle formula to solve, 𝐬𝐢𝐧 𝒙 𝟐 =𝟏−𝐜𝐨𝐬𝒙
5/20/2019 6:17 PM
9.3A: Double Angles Identity

21. 9.3A: Double Angles Identity
Assignment
Worksheet
5/20/2019 6:17 PM
9.3A: Double Angles Identity