4.7 Inverse Trig Functions

Objective

• Evaluate and graph the inverse sine, cosine and tangent function.

• Evaluate the compositions of trig functions

Inverse Sine Function

y

2

1

1

x

y = sin x

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 does not pass the Horizontal Line Test

and must be restricted to find its inverse.

Take a look at the piece of the graph in the red frame.

x

yIf we restrict the domain to the interval [-π/2, π/2]. The following properties hold:

1. On the interval [-π/2, π/2], the function y = sin x is increasing.

2. On the interval [-π/2, π/2],y = sin x takes on its full range of values, [-1, 1].

Lets zoom in and look at some key points in this section.

x

yy = sin(x)

3. On the interval [-π/2, π/2],y = sin x is one-to-one.

–1.5 1.5

–

The inverse sine function is defined byy = arcsin x if and only if sin y = x.

Angle whose sine is x

The domain of y = arcsin x is [–1, 1].

Example:

1a. arcsin2 6

1 is the angle whose sine is .6 2

1 3b. sin2 3

3sin3 2

This is another way to write arcsin x.

The range of y = arcsin x is [–/2 , /2].

Inverse Cosine Function

Cos x has an inverse function on this interval.

f(x) = cos x must be restricted to find its inverse.

The cosine function is decreasing and one-to-one on the interval [0, π]

y

2

1

1

x

y = cos x

x

yy = arccos(x)

The inverse cosine function is defined byy = arccos x if and only if cos y = x.

Angle whose cosine is x

The domain of y = arccos x is [–1, 1].

Example: 1a.) arccos2 3

1 is the angle whose cosine is .3 2

1 3 5b.) cos2 6

35cos6 2

This is another way to write arccos x.

The range of y = arccos x is [0 , ].

Inverse Tangent Functionf(x) = tan x must be restricted to find its inverse.

Tan x has an inverse function on this interval.

y

x

2

3

2

32

2

y = tan x

x

y

The inverse tangent function is defined byy = arctan x if and only if tan y = x.

Angle whose tangent is x

Example: 3a.) arctan

3 6 3 is the angle whose tangent is .

6 3

1b.) tan 33 tan 3

3

This is another way to write arctan x.

The domain of y = arctan x is .( , ) The range of y = arctan x is (–/2 , /2).

Composition of Functions:f(f –1(x)) = x and (f –1(f(x)) = x.

If –1 x 1 and – /2 y /2, thensin(arcsin x) = x and arcsin(sin y) = y.

If –1 x 1 and 0 y , thencos(arccos x) = x and arccos(cos y) = y.

If x is a real number and –/2 < y < /2, thentan(arctan x) = x and arctan(tan y) = y.

Example: tan(arctan 4) = 4

Inverse Properties:

Example:

a. sin–1(sin (–/2)) = –/2

1 5b. sin sin3

53 does not lie in the range of the arcsine function, –/2 y /2.

y

x

53

3

5 23 3 However, it is coterminal with

which does lie in the range of the

arcsine function.

1 15sin sin sin sin3 3 3

Example:

2Find the exact value of tan arccos .3

x

y

3

2

adj2 2Let = arccos , then cos .3 hyp 3

u u

2 23 2 5

opp 52tan arccos tan3 adj 2

u

u