1. 7.1: Graphs of Sin, Cos, and Tan
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7.1: Graphs of Sin, Cos, and Tan

2. Inverse Functions Section 8.2
Pre–Calculus AB PreAP/Dual, Revised ©2014
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7.1: Graphs of Sin, Cos, and Tan

3. 7.1: Graphs of Sin, Cos, and Tan
Definitions
The result of exchanging the input and output value of a relation is an Inverse Function
An inverse “undoes” the function. It switches (x, y) to (y, x)
Interchange the x and the y. (make y x and make x y)
Written in function notation as f –1(x)
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7.1: Graphs of Sin, Cos, and Tan

4. 7.1: Graphs of Sin, Cos, and Tan
Review Example 1
Determine the inverse of this relation, {(0, –3), (2, 1), and (6, 3)}
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7.1: Graphs of Sin, Cos, and Tan

5. 6.4 – Use Inverse Functions
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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6.4 – Use Inverse Functions

6. Review Example 1 inverse Mirrored Image y = x 11/18/2018 4:02 PM
5.3 – Inverses

7. To find f’(x), we have the HORIZONTAL LINE TEST
Question
If a function is a relation, is an inverse a function as well?
inverse NO
REPEATING
Y’s
To find f’(x), we have the HORIZONTAL LINE TEST
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5.3 – Inverses

8. 6.4 – Use Inverse Functions
Review Example 2
Determine the inverse of y = 3x – 2
y = 3x – 2 y
= 3 – 2 x
x = 3y – 2
x + 2 = 3y
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6.4 – Use Inverse Functions

9. 7.1: Graphs of Sin, Cos, and Tan
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–1a = θ, where Sin θ = a
There are two ways of writing inverse functions for sine
SIN–1
arcsin
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7.1: Graphs of Sin, Cos, and Tan

10. The Inverse of Sine Graph
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7.1: Graphs of Sin, Cos, and Tan

11. 7.1: Graphs of Sin, Cos, and Tan
Sine Function
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7.1: Graphs of Sin, Cos, and Tan

12. 7.1: Graphs of Sin, Cos, and Tan
The Inverse of Sine x y
π/2 1 π
3π/2 –1 2π x y 1
π/2 π –1
3π/2 2π
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7.1: Graphs of Sin, Cos, and Tan

13. 7.1: Graphs of Sin, Cos, and Tan
Cosine Function
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7.1: Graphs of Sin, Cos, and Tan

14. The Inverse of Cosine Graph
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7.1: Graphs of Sin, Cos, and Tan

15. 7.1: Graphs of Sin, Cos, and Tan
The Inverse of Cosine x y 1
π/2 π –1
3π/2 2π x y 1
π/2 –1 π
3π/2 2π
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7.1: Graphs of Sin, Cos, and Tan

16. The Inverse of Tangent Graph
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7.1: Graphs of Sin, Cos, and Tan

17. 7.1: Graphs of Sin, Cos, and Tan
Tangent Function
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7.1: Graphs of Sin, Cos, and Tan

18. 7.1: Graphs of Sin, Cos, and Tan
The Inverse of Tangent x y
Und
π/2 –1 π
3π/2 1 2π x y
Und –1
π/2 π 1
3π/2 2π
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7.1: Graphs of Sin, Cos, and Tan

19. 7.1: Graphs of Sin, Cos, and Tan
The Inverse Graphs
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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7.1: Graphs of Sin, Cos, and Tan

20. 7.1: Graphs of Sin, Cos, and Tan
Review
Find the exact value of cos(3π/4)
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7.1: Graphs of Sin, Cos, and Tan

21. 7.1: Graphs of Sin, Cos, and Tan
Example 1
Find the exact value of 𝐂𝐨𝐬 −𝟏 − 𝟐 𝟐
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7.1: Graphs of Sin, Cos, and Tan

22. 7.1: Graphs of Sin, Cos, and Tan
Example 2
Find the exact value of 𝐒𝐢𝐧 −𝟏 𝟏 𝟐
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7.1: Graphs of Sin, Cos, and Tan

23. 7.1: Graphs of Sin, Cos, and Tan
Example 3
Find the exact value of 𝒂𝒓𝒄𝒄𝒐𝒔 − 𝟐 𝟐
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7.1: Graphs of Sin, Cos, and Tan

24. 7.1: Graphs of Sin, Cos, and Tan
Your Turn
Find the exact value of 𝐂𝐨𝐬 −𝟏 𝟐 𝟐
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7.1: Graphs of Sin, Cos, and Tan

25. 7.1: Graphs of Sin, Cos, and Tan
Example 4
Find the exact value of 𝐓𝐚𝐧 −𝟏 𝟑
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7.1: Graphs of Sin, Cos, and Tan

26. 7.1: Graphs of Sin, Cos, and Tan
Example 5
Find the exact value of 𝒂𝒓𝒄𝒔𝒊𝒏 − 𝟑 𝟐
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7.1: Graphs of Sin, Cos, and Tan

27. 7.1: Graphs of Sin, Cos, and Tan
Your Turn
Find the exact value of 𝐓𝐚𝐧 −𝟏 − 𝟑 𝟑
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7.1: Graphs of Sin, Cos, and Tan

28. 7.1: Graphs of Sin, Cos, and Tan
Assignment
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7.1: Graphs of Sin, Cos, and Tan