1. Inverse Trig Functions
Section 5.6
Calculus AP/Dual, Revised ©2018
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§5.6: Inverse Trig Functions

2. §5.6: Inverse Trig Functions
Review
What is the domain and range of 𝒇 𝒙 =𝐬𝐢𝐧 −𝟏 𝒙 , 𝐜𝐨𝐬 −𝟏 𝒙 , 𝐚𝐧𝐝 𝐭𝐚𝐧 −𝟏 𝒙 ?
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§5.6: Inverse Trig Functions

3. §5.6: Inverse Trig Functions
Inverse Sine
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§5.6: Inverse Trig Functions

4. §5.6: Inverse Trig Functions
Inverse Cosine
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§5.6: Inverse Trig Functions

5. §5.6: Inverse Trig Functions
Inverse Tangent
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§5.6: Inverse Trig Functions

6. §5.6: Inverse Trig Functions
Sine Function
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§5.6: Inverse Trig Functions

7. §5.6: Inverse Trig Functions
Cosine Function
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§5.6: Inverse Trig Functions

8. §5.6: Inverse Trig Functions
Tangent Function
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§5.6: Inverse Trig Functions

9. The Inverse Graph Restrictions
The restricted domain for sine function is from − 𝝅 𝟐 , 𝝅 𝟐 (Quad I and IV)
The restricted domain for cosine function is from 𝟎,𝝅 (Quad I and II)
The restricted domain for tangent function is from − 𝝅 𝟐 , 𝝅 𝟐 (Quad I and IV excluding − 𝝅 𝟐 & 𝝅 𝟐 )
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§5.6: Inverse Trig Functions

10. §5.6: Inverse Trig Functions
Inverses
None of the six basic trig functions have inverse functions.
Restrictions of the domain allow them to have inverses, as demonstrated with the horizontal line test
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§5.6: Inverse Trig Functions

11. §5.6: Inverse Trig Functions
Example 1
Without a calculator, solve 𝐚𝐫𝐜𝐜𝐨𝐬 − 𝟑 𝟐 .
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§5.6: Inverse Trig Functions

12. §5.6: Inverse Trig Functions
Your Turn
Without a calculator, solve 𝐚𝐫𝐜𝐬𝐢𝐧 − 𝟐 𝟐 .
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§5.6: Inverse Trig Functions

13. §5.6: Inverse Trig Functions
Example 2
Evaluate 𝐚𝐫𝐜𝐬𝐢𝐧 −𝟎.𝟑𝟗
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§5.6: Inverse Trig Functions

14. §5.6: Inverse Trig Functions
Example 3
Without a calculator, solve 𝐜𝐨𝐬 𝐬𝐢𝐧 −𝟏 − 𝟐 𝟐 .
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§5.6: Inverse Trig Functions

15. §5.6: Inverse Trig Functions
Example 4
Without a calculator, solve 𝐜𝐨𝐬 𝐬𝐢𝐧 −𝟏 𝟓 𝟏𝟑 .
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§5.6: Inverse Trig Functions

16. §5.6: Inverse Trig Functions
Example 5
Without a calculator, solve 𝐬𝐞𝐜 𝐭𝐚𝐧 −𝟏 𝟒𝒙 .
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§5.6: Inverse Trig Functions

17. §5.6: Inverse Trig Functions
Your Turn
Without a calculator, solve 𝐜𝐨𝐬 𝐜𝐨𝐭 −𝟏 𝒙 .
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§5.6: Inverse Trig Functions

18. §5.6: Inverse Trig Functions
Proof
Take the derivative of 𝒚= 𝐬𝐢𝐧 −𝟏 𝒙 .
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§5.6: Inverse Trig Functions

19. §5.6: Inverse Trig Functions
Proof
Take the derivative of 𝒚= 𝐬𝐢𝐧 −𝟏 𝒙 .
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§5.6: Inverse Trig Functions

20. Derivative of Inverse Trig Functions
𝒅 𝒅𝒙 𝐬𝐢𝐧 −𝟏 𝒖 = 𝒅𝒖 𝟏− 𝒖 𝟐
𝒅 𝒅𝒙 𝐜𝐨𝐬 −𝟏 𝒖 =− 𝒅𝒖 𝟏− 𝒖 𝟐
𝒅 𝒅𝒙 𝐭𝐚𝐧 −𝟏 𝒖 = 𝒅𝒖 𝟏+ 𝒖 𝟐
𝒅 𝒅𝒙 𝐜𝐨𝐭 −𝟏 𝒖 =− 𝒅𝒖 𝟏+ 𝒖 𝟐
𝒅 𝒅𝒙 𝐬𝐞𝐜 −𝟏 𝒖 = 𝒅𝒖 𝒖 𝒖 𝟐 −𝟏
𝒅 𝒅𝒙 𝐜𝐬𝐜 −𝟏 𝒖 =− 𝒅𝒖 𝒖 𝒖 𝟐 −𝟏
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§5.6: Inverse Trig Functions

21. §5.6: Inverse Trig Functions
Example 6
Solve 𝒅 𝒅𝒕 𝐬𝐢𝐧 −𝟏 𝒕 𝟐
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§5.6: Inverse Trig Functions

22. §5.6: Inverse Trig Functions
Example 7
Solve 𝒅 𝒅𝒙 𝐚𝐫𝐜𝐭𝐚𝐧 𝟑𝒙
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§5.6: Inverse Trig Functions

23. §5.6: Inverse Trig Functions
Example 8
Solve 𝒅 𝒅𝒙 𝟐 𝐬𝐞𝐜 −𝟏 𝒆 𝟐𝒙
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§5.6: Inverse Trig Functions

24. §5.6: Inverse Trig Functions
Your Turn
Solve 𝒅 𝒅𝒙 𝐚𝐫𝐜𝐜𝐨𝐬 𝒙 𝟐
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§5.6: Inverse Trig Functions

25. §5.6: Inverse Trig Functions
Example 9
An equation for a tangent to the graph of 𝒚=𝐚𝐫𝐜𝐬𝐢𝐧 𝒙 𝟐 at the origin.
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§5.6: Inverse Trig Functions

26. §5.6: Inverse Trig Functions
Example 9
An equation for a tangent to the graph of 𝒚=𝐚𝐫𝐜𝐬𝐢𝐧 𝒙 𝟐 at the origin.
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§5.6: Inverse Trig Functions

27. AP Multiple Choice Practice Question 1 (non-calculator)
Solve 𝒅 𝒅𝒙 𝐚𝐫𝐜𝐬𝐢𝐧 𝟐𝒙 (A) −𝟏 𝟐 𝟏−𝟒 𝒙 𝟐 (B) −𝟐 𝟒 𝒙 𝟐 −𝟏 (C) 𝟏 𝟐 𝟏−𝟒 𝒙 𝟐 (D) 𝟐 𝟏−𝟒 𝒙 𝟐
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§5.6: Inverse Trig Functions

28. AP Multiple Choice Practice Question 1 (non-calculator)
Solve 𝒅 𝒅𝒙 𝐚𝐫𝐜𝐬𝐢𝐧 𝟐𝒙
Vocabulary
Connections and Process
Answer and Justifications
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§5.6: Inverse Trig Functions

29. §5.6: Inverse Trig Functions
Assignment
Page 372
3-19 EOO, odd, odd
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§5.6: Inverse Trig Functions