Chapter 5 – Trigonometric Functions: Unit Circle Approach 5.5 - Inverse Trigonometric Functions &...

5.5 - Inverse Trigonometric Functions & Their Graphs

Section 5.5 Inverse

Trigonometric Functions & Their Graphs

Chapter 5 – Trigonometric Functions: Unit Circle Approach


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.


Sine FunctionDoes not pass the horizontal line test.Must restrict the domain to create an inverse

function.


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


Note


Graph of Inverse sine


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


ExamplesFind the exact value of the following:

1

21. arcsin

2

32. sin

2 1

13. arcsin

2

4. sin 2


Cosine FunctionDoes not pass the horizontal line test.Must restrict the domain to create an inverse

function.


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


Graph of Inverse Cosine


Cancellation Properties - Cosine

Thus y = cos-1x is the number in the interval [0, ] whose cosine is x.


1

1

cos cos for 1 1

cos cos for 0

x x x

x x x



1

21. arccos

2

32. cos

2 1

13. arccos

2

34. cos

2


Tangent FunctionDoes not pass the horizontal line test.Must restrict the domain to create an inverse

function.


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


Graph of Inverse Tangent


Cancellation Properties - Tangent

Thus y = tan-1x is the number in the interval (- ⁄ 2, ⁄ 2) whose sine is x.


1

1

tan tan for

tan tan for 2 2

x x x

x x x



1

1. arctan 3

2. tan 1


Evaluating Compositions


Inverse Properties


Using Inverse Properties

Evaluate the following:

1

1

3. tan tan 5

4. cos cos

1

1

1. sin sin4

72. sin sin

4


Examples – pg. 412Find the exact value of the expression if it is defined.

1 1

1 1

139. tan sin 40. cos sin 0

2

3 241. cos sin 42. tan sin

2 2

Chapter 5 – Trigonometric Functions: Unit Circle Approach 5.5 - Inverse Trigonometric Functions &...

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