1. Inverse Trigonometric Functions

2. Graph y = sinx Graph y = -1/2 Domain: x is element of all reals
Range: {y| -1 ≤ y ≤ 1}
Graph y = -1/2
Where do these graphs cross?
Is y = sinx one-to-one?

3. Inverse Sine Function Sin x has an inverse function on this interval.
Recall that for a function to have an inverse, it must be a one-to-one function and pass the Horizontal Line Test.
f(x) = sin x is not one-to-one
and must be restricted to find its inverse. y x
y = sin x
Sin x has an inverse function on this interval.
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Inverse Sine Function

4. The inverse sine function is defined by
y = Arcsin x if and only if Sin y = x.
Angle whose sine is x
The domain of y = Arcsin x is {x | –1≤ x ≤ 1}
The range of y = Arcsin x is { y | –/2 ≤ y ≤ /2}
Example:
This is another way to write Arcsin x.
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Inverse Sine Function

5. Inverse Cosine Function
f(x) = cos x must be restricted to find its inverse. y x
y = cos x
Cos x has an inverse function on this interval.
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Inverse Cosine Function

6. Inverse Cosine Function
The inverse cosine function is defined by
y = Arccos x if and only if Cos y = x.
Angle whose cosine is x
The domain of y = Arccos x is { x | –1 ≤ x ≤ 1}
The range of y = Arccos x is { y | 0 ≤ y ≤ }
Example:
This is another way to write Arccos x.
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Inverse Cosine Function

7. Inverse Tangent Function
f(x) = tan x must be restricted to find its inverse. y x
y = tan x
Tan x has an inverse function on this interval.
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Inverse Tangent Function

8. Inverse Tangent Function
The inverse tangent function is defined by
y = Arctan x if and only if Tan y = x.
Angle whose tangent is x
The domain of y = Arctan x is all real numbers
The range of y = Arctan x is { y | –/2 ≤ y ≤ /2}
Example:
This is another way to write Arctan x.
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Inverse Tangent Function

9. Graphing Utility: Graphs of Inverse Functions
Graphing Utility: Graph the following inverse functions.
Set calculator to radian mode.
–1.5
1.5 – 
a. y = Arcsin x
–1.5
1.5 2 –
b. y = Arccos x –3 3  –
c. y = Arctan x
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Graphing Utility: Graphs of Inverse Functions

10. Graphing Utility: Inverse Functions
Graphing Utility: Approximate the value of each expression.
Set calculator to radian mode.
a. Cos–1 0.75
b. Arcsin 0.19
Why?
c. Arctan 1.32
d. Arcsin 2.5
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Graphing Utility: Inverse Functions

11. Composition of Functions
Recall  f(f –1(x)) = x and (f –1(f(x)) = x.
Inverse Properties:
If –1  x  1 and – /2  y  /2, then
sin(Arcsin x) = x and Arcsin(sin y) = y.
If –1  x  1 and 0  y  , then
cos(Arccos x) = x and Arccos(cos y) = y.
If x is a real number and –/2 < y < /2, then
tan(Arctan x) = x and Arctan(tan y) = y.
Example: tan(Arctan 4) = 4
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Composition of Functions

12. Composition of Functions
Example:
a. Sin–1(sin (–/2)) = –/2
does not lie in the range of the Arcsine function, –/2  y  /2.
However, it is coterminal with which does lie in the range of the Arcsine function. y x
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Composition of Functions

13. Example: Evaluating Composition of Functionsy 3 u 2
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Example: Evaluating Composition of Functions

14. Example: Evaluating Composition of Functions
OR… y 2 x 3
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Example: Evaluating Composition of Functions