1. Inverse Trig Functions
Section 4.7
Precalculus PreAP/Dual, Revised ©2017
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§4.7: Inverse Functions

2. {( , ), ( , ), ( , )} –3 2 1 6 3 Review Example 1
Determine the inverse of this relation, {(0, –3), (2, 1), and (6, 3)}
{( , ), ( , ), ( , )} –3 2 1 6 3
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§4.7: Inverse Functions

3. Review Example 1 𝒚=𝒙 inverse Mirrored Image 5/27/2019 10:40 PM
§4.7: Inverse Functions

4. To find 𝒇 −𝟏 (𝒙), we have the HORIZONTAL LINE TEST
Question
If a function is a relation, is an inverse a function as well?
inverse
To find 𝒇 −𝟏 (𝒙), we have the HORIZONTAL LINE TEST
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§4.7: Inverse Functions

5. …to Solve the inverse, switch the 𝒙’s and 𝒚’s
Review Example 2
Determine the inverse of 𝒚=𝟑𝒙−𝟐
…to Solve the inverse, switch the 𝒙’s and 𝒚’s
y = 3x – 2 y
= 3 – 2 x
x = 3y – 2
x + 2 = 3y
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§4.7: Inverse Functions

6. Inverse Definitions 𝑺𝒊𝒏 −𝟏 is read as “the inverse sine”
Indicates the inverse of the sine function
Cosine’s inverse is 𝑪𝒐𝒔 −𝟏 , etc…
𝑺𝒊𝒏 −𝟏 𝒂=𝜽, where 𝐒𝐢𝐧 𝜽=𝒂
There are two ways of writing inverse functions for sine
𝑺𝒊𝒏 −𝟏
𝒂𝒓𝒄𝒔𝒊𝒏
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§4.7: Inverse Functions

7. The Inverse of Sine Graph
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§4.7: Inverse Functions

8. Sine Function
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9. The Inverse of Sine 𝒙 𝒚 𝟎 𝝅/𝟐 𝟏 𝝅 𝟑𝝅/𝟐 −𝟏 𝟐𝝅 𝒙 𝒚 𝟎 𝟏 𝝅/𝟐 𝝅 −𝟏 𝟑𝝅/𝟐 𝟐𝝅
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§4.7: Inverse Functions

10. Cosine Function
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11. The Inverse of Cosine Graph
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12. The Inverse of Cosine 𝒙 𝒚 𝟎 𝟏 𝝅/𝟐 𝝅 −𝟏 𝟑𝝅/𝟐 𝟐𝝅 𝒙 𝒚 𝟏 𝟎 𝝅/𝟐 −𝟏 𝝅 𝟑𝝅/𝟐
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§4.7: Inverse Functions

13. The Inverse of Tangent Graph
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14. Tangent Function
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15. The Inverse of Tangent 𝒙 𝒚 𝟎 𝑼𝒏𝒅 𝝅/𝟐 −𝟏 𝝅 𝟑𝝅/𝟐 𝟏 𝟐𝝅 𝒙 𝒚 𝑼𝒏𝒅 𝟎 −𝟏 𝝅/𝟐 𝝅
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§4.7: Inverse Functions

16. The Inverse Graph Restrictions
The restricted domain for sine function is from − 𝝅 𝟐 , 𝝅 𝟐
The restricted domain for cosine function is from 𝟎,𝝅
The restricted domain for tangent function is from − 𝝅 𝟐 , 𝝅 𝟐
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§4.7: Inverse Functions

17. Review Find the exact value of 𝐜𝐨𝐬 𝟑𝝅 𝟒 5/27/2019 10:40 PM
§4.7: Inverse Functions

18. Example 1 Find the exact value of 𝐂𝐨𝐬 −𝟏 − 𝟐 𝟐 5/27/2019 10:40 PM
§4.7: Inverse Functions

19. Example 2 Find the exact value of 𝐚𝐫𝐜𝐬𝐢𝐧 𝟏 𝟐 5/27/2019 10:40 PM
§4.7: Inverse Functions

20. Your Turn Find the exact value of 𝐂𝐨𝐬 −𝟏 𝟐 𝟐 5/27/2019 10:40 PM
§4.7: Inverse Functions

21. Example 3 Find the exact value of 𝐓𝐚𝐧 −𝟏 𝟑 5/27/2019 10:40 PM
§4.7: Inverse Functions

22. Example 4 Find the exact value of 𝐚𝐫𝐜𝐬𝐢𝐧 − 𝟑 𝟐 5/27/2019 10:40 PM
§4.7: Inverse Functions

23. Your Turn Find the exact value of 𝐓𝐚𝐧 −𝟏 − 𝟑 𝟑 5/27/2019 10:40 PM
§4.7: Inverse Functions

24. The Inverse of Cosecant Graph
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§4.7: Inverse Functions

25. Cosecant Function
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26. The Inverse of Secant Graph
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27. Secant Function
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28. The Inverse of Cotangent Graph
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29. Cotangent Function
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30. The Inverse Reciprocal Graphs
The restricted domain for cosecant function is from − 𝝅 𝟐 , 𝝅 𝟐
The restricted domain for secant function is from 𝟎,𝝅
The restricted domain for cotangent function is from 𝟎,𝝅
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§4.7: Inverse Functions

31. Steps Take the reciprocal of the trig function
Take the reciprocal of the answer
Switch and solve
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§4.7: Inverse Functions

32. Example 5 Find the exact value of 𝑺𝒆𝒄 −𝟏 − 𝟐 5/27/2019 10:40 PM
§4.7: Inverse Functions

33. Example 6 Find the exact value of 𝐚𝐫𝐜𝐬𝐞𝐜 − 𝟐 𝟑 𝟑 5/27/2019 10:40 PM
§4.7: Inverse Functions

34. Example 7 Find the exact value of 𝑪𝒐𝒕 −𝟏 − 𝟑 5/27/2019 10:40 PM
§4.7: Inverse Functions

35. Your Turn Find the exact value of 𝑪𝒔𝒄 −𝟏 − 𝟐 5/27/2019 10:40 PM
§4.7: Inverse Functions

36. Inverses in the Calculator
Make sure that it is in the appropriate mode
It is the 𝑻𝒓𝒊𝒈 −𝟏 feature in the graphing calculator
For reciprocal functions, type in the 𝑻𝒓𝒊𝒈 −𝟏 𝟏 𝒂𝒏𝒔
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§4.7: Inverse Functions

37. Example 8 Use the calculator to approximate 𝐚𝐫𝐜𝐭𝐚𝐧(−𝟖.𝟒𝟓)
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§4.7: Inverse Functions

38. Example 9 Use the calculator to approximate 𝑪𝒐𝒕 −𝟏 𝟑
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§4.7: Inverse Functions

39. Your Turn Use the calculator to approximate 𝒂𝒓𝒄𝒄𝒔𝒄 −𝟏𝟐
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§4.7: Inverse Functions

40. Assignment
Page odd (without a calculator), EOO (with a calculator), all (without), all (Round for 4 decimal places)
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§4.7: Inverse Functions