4.7 Inverse Trigonometric Functions

Inverse functions

• g(x) is the inverse function of f(x) IFg(f(x)) = x and f(g(x)) = x

We notate an inverse function as f-1(x)

Example f(x) = 4x f-1(x)= 4

x

Remember your favorite inverse functions?

Logarithms and exponentials?

f(x) = 2x f-1(x)= log2x

Starboard demo

Does it pass the vertical line test? Yes, so it’s a function!!

Does it pass the horizontal line test?

NOOO!!It does not have an inverse function

Restrict the domain to f(x)= x2 , x ≥ 0

Now it passes the horizontal line test.

Consider the graph f(x) = sinx

Is it one-to-one?

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

IT MUST BE RESTRICTED!!

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

Cartoon time

• Oh, sine machine. He is soo happy outputting side to side ratios….Takes in angles- outputs side to side ratios…

BUT, when his arch enemy ARCSINE comes along, he has to fight the guy who undoes everything he does.

Inverse Cosine Function

Cos x has an inverse function on this interval.

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

y

2

1

1

x

y = cos 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

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].

Examples

Consider a slightly different setup:

120sinarcsin This is also the composition of two inverse functions but…

.602

3arcsin

Did you suspect the answer was going to be 120 degrees? This problem behaved differently because the first angle, 120 degrees, was outside the range of the arcsin. So use some caution when evaluating the composition of inverse trig functions.

The remainder of this presentation consists of practice problems, their answers and a few complete solutions.

Find the six trig functions of Ө=30o Ө 2

1

2

3

1

Warm-up

301

2

3

2

Triangle A

Triangle B

• Page 328 #1-4; 6-8; 13-20; 37; 40; 49-55; 71

Restricted domain

• How to tell if a function has an inverse function

Examples

For a function to have an inverse function, it has to be one-to-one

Does it pass the vertical line test? Yes, so it’s a function!!

Does it pass the horizontal line test?

NOOO!!It does not have an inverse function

X -1 1 0 -2 2y 1 1 0 4 4

X 1 1 0 4 4y -1 1 0 -2 2