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5.5 - Inverse Trigonometric Functions & Their Graphs
Section 5.5 Inverse
Trigonometric Functions & Their Graphs
Chapter 5 – Trigonometric Functions: Unit Circle Approach
5.5 - Inverse Trigonometric Functions & Their Graphs
Review of Inverse Functions
RememberIf the graph passes the horizontal line test, then the
function has an inverse functions.
If a point (a, b) is on the graph of f, then the point (b, a) is on the graph of f -1.
The graph of f -1 is a reflection of the graph of f about the line y=x.
5.5 - Inverse Trigonometric Functions & Their Graphs
Sine FunctionDoes not pass the horizontal line test.Must restrict the domain to create an inverse
function.
5.5 - Inverse Trigonometric Functions & Their Graphs
DefinitionThe inverse sine function is the function sin-1 with
domain [-1, 1] and range [- ⁄ 2, ⁄ 2] defined by
The inverse sine function is also called arcsine denoted by arcsin.
1sin sinx y y x
5.5 - Inverse Trigonometric Functions & Their Graphs
Cancellation Properties - SineThus y = sin-1x is the number in the interval
[- ⁄ 2, ⁄ 2] whose sine is x.
In other words we have the following:
1
1
sin sin for 1 1
sin sin for 2 2
x x x
x x x
5.5 - Inverse Trigonometric Functions & Their Graphs
ExamplesFind the exact value of the following:
1
21. arcsin
2
32. sin
2 1
13. arcsin
2
4. sin 2
5.5 - Inverse Trigonometric Functions & Their Graphs
Cosine FunctionDoes not pass the horizontal line test.Must restrict the domain to create an inverse
function.
5.5 - Inverse Trigonometric Functions & Their Graphs
DefinitionThe inverse cosine function is the function cos-1
with domain [-1, 1] and range [0, ] defined by
The inverse sine function is also called arccosine denoted by arccos.
1cos cosx y y x
5.5 - Inverse Trigonometric Functions & Their Graphs
Cancellation Properties - Cosine
Thus y = cos-1x is the number in the interval [0, ] whose cosine is x.
In other words we have the following:
1
1
cos cos for 1 1
cos cos for 0
x x x
x x x
5.5 - Inverse Trigonometric Functions & Their Graphs
ExamplesFind the exact value of the following:
1
21. arccos
2
32. cos
2 1
13. arccos
2
34. cos
2
5.5 - Inverse Trigonometric Functions & Their Graphs
Tangent FunctionDoes not pass the horizontal line test.Must restrict the domain to create an inverse
function.
5.5 - Inverse Trigonometric Functions & Their Graphs
DefinitionThe inverse tangent function is the function tan-1
with domain (-∞, ∞) and range (- ⁄ 2, ⁄ 2) defined by
The inverse tangent function is also called arctangent denoted by arctan.
1tan tanx y y x
5.5 - Inverse Trigonometric Functions & Their Graphs
Cancellation Properties - Tangent
Thus y = tan-1x is the number in the interval (- ⁄ 2, ⁄ 2) whose sine is x.
In other words we have the following:
1
1
tan tan for
tan tan for 2 2
x x x
x x x
5.5 - Inverse Trigonometric Functions & Their Graphs
ExamplesFind the exact value of the following:
1
1. arctan 3
2. tan 1
5.5 - Inverse Trigonometric Functions & Their Graphs
Using Inverse Properties
Evaluate the following:
1
1
3. tan tan 5
4. cos cos
1
1
1. sin sin4
72. sin sin
4