1. Precalculus PreAP/Dual, Revised ©2017 §5.5: Double Angles Identity
Section 5.5
Precalculus PreAP/Dual, Revised ©2017
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§5.5: Double Angles Identity

2. Double Angle Identities (fill in chart)
𝐬𝐢𝐧 𝟐𝜽=
𝟐 𝐬𝐢𝐧𝜽 𝐜𝐨𝐬𝜽
𝐜𝐨𝐬 𝟐𝜽=
𝐜𝐨𝐬 𝟐 𝜽− 𝐬𝐢𝐧 𝟐 𝜽
𝟐 𝐜𝐨𝐬 𝟐 𝜽−𝟏
𝟏−𝟐 𝐬𝐢𝐧 𝟐 𝜽
𝐭𝐚𝐧 𝟐𝜽=
𝟐 𝐭𝐚𝐧𝜽 𝟏− 𝐭𝐚𝐧 𝟐 𝜽
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§5.5: Double Angles Identity

3. §5.5: Double Angles Identity
Proof
Proof of 𝐬𝐢𝐧 𝟐𝜽
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§5.5: Double Angles Identity

4. §5.5: Double Angles Identity
Proof
Proof of 𝐜𝐨𝐬 𝟐𝜽
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§5.5: Double Angles Identity

5. §5.5: Double Angles Identity
Proof
Proof of 𝐜𝐨𝐬 𝟐𝜽 (Second Part)
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§5.5: Double Angles Identity

6. §5.5: Double Angles Identity
Proof
Proof of 𝐜𝐨𝐬 𝟐𝜽 (Third Part)
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§5.5: Double Angles Identity

7. §5.5: Double Angles Identity
Proof
Proof of 𝐭𝐚𝐧 𝟐𝜽
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§5.5: Double Angles Identity

8. Steps for Double Angle Identity
Simplify the equation
Substitute any identities with a double angle identity
Factoring
𝑼-Substitution for Trigonometry
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
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§5.5: Double Angles Identity

9. §5.5: Double Angles Identity
Example 1
Use the double angle formula to solve, 𝐬𝐢𝐧 𝟐𝒙−𝐬𝐢𝐧𝒙=𝟎
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§5.5: Double Angles Identity

10. §5.5: Double Angles Identity
Example 1
Use the double angle formula to solve, 𝐬𝐢𝐧 𝟐𝒙−𝐬𝐢𝐧𝒙=𝟎
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§5.5: Double Angles Identity

11. §5.5: Double Angles Identity
Example 2
Use the double angle formula to solve, 𝐬𝐢𝐧 𝟐𝒙+𝐜𝐨𝐬 𝒙=𝟎
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§5.5: Double Angles Identity

12. §5.5: Double Angles Identity
Your Turn
Use the double angle formula to solve, 𝟐𝐜𝐨𝐬 𝒙+𝐬𝐢𝐧 𝟐𝒙=𝟎
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§5.5: Double Angles Identity

13. §5.5: Double Angles Identity
Example 3
Use the double angle formula to solve, 𝒚=𝟒 𝐜𝐨𝐬 𝟐 𝒙−𝟐
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§5.5: Double Angles Identity

14. §5.5: Double Angles Identity
Example 3
Use the double angle formula to solve, 𝒚=𝟒 𝐜𝐨𝐬 𝟐 𝒙−𝟐
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§5.5: Double Angles Identity

15. §5.5: Double Angles Identity
Example 3
Use the double angle formula to solve, 𝒚=𝟒 𝐜𝐨𝐬 𝟐 𝒙−𝟐
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§5.5: Double Angles Identity

16. §5.5: Double Angles Identity
Example 3
Use the double angle formula to solve, 𝒚=𝟒 𝐜𝐨𝐬 𝟐 𝒙−𝟐
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§5.5: Double Angles Identity

17. §5.5: Double Angles Identity
Example 4
Use the double angle formula to solve, 𝟑 𝐬𝐢𝐧 𝟐 𝒙+𝐜𝐨𝐬 𝟐𝒙−𝟐=𝟎
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§5.5: Double Angles Identity

18. §5.5: Double Angles Identity
Your Turn
Use the double angle formula to solve, 𝐜𝐨𝐬 𝟐𝒙+𝐜𝐨𝐬 𝒙=𝟎
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§5.5: Double Angles Identity

19. §5.5: Double Angles Identity
Example 5
Given 𝐬𝐢𝐧 𝒙= 𝟒 𝟓 from 𝝅 𝟐 ≤𝒙≤𝝅 identify the value of 𝐬𝐢𝐧 𝟐𝒙 .
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§5.5: Double Angles Identity

20. §5.5: Double Angles Identity
Example 6
Given 𝐬𝐢𝐧 𝒙=− 𝟕 𝟐𝟓 from 𝝅≤𝒙≤ 𝟑𝝅 𝟐 identify the value of 𝐜𝐨𝐬 𝟐𝒙 .
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§5.5: Double Angles Identity

21. §5.5: Double Angles Identity
Your Turn
Given 𝐜𝐨𝐬 𝜽= 𝟓 𝟏𝟑 from 𝟑𝝅 𝟐 ≤𝒙≤𝟐𝝅 identify the value of 𝐭𝐚𝐧 𝟐𝜽 .
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§5.5: Double Angles Identity

22. §5.5: Double Angles Identity
Assignment
Worksheet
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§5.5: Double Angles Identity