4.7Inverse Trig Functions

JMerrill, 2010

We know that for a function to have an inverse function, it must be one-to-one (it must pass the Horizontal Line Test).

Recall

From looking at a sine wave, it is obvious that it does not pass the Horizontal Line Test.

Sine Wave

The Unit Circle in found in section 4.7. We will use:

◦ Radians◦ Exact answers (mostly)◦ Quick board review of Unit Circle, quadrants

on the wave, & converting to radian measure

The Unit Circle

In order to pass the Horizontal Line Test (so that sin x has an inverse that is a function), we must restrict the domain.

We restrict it to

Sine Wave

,2 2

Quadrant IV is Quadrant I is Answers must be in one of those two

quadrants or the answer doesn’t exist.

Sine Wave,0

2

0,

2

How do we draw inverse functions? Switch the x’s and y’s!

Sine Wave

Switching the x’s and y’s also means switching the axis!

Domain/range of restricted sine wave? Domain/range of inverse?

Sine Wave

: ,2 2

: 1,1

D

R

: 1,1

: ,2 2

D

R

y = arcsin x or y = sin-1 x

Both mean the same thing. They mean that you’re looking for the angle where sin y = x.

Inverse Notation

Find the exact value of: Arcsin ½

◦ This means at what angle is the sin = ½ ?◦ π/6◦ (5π/6 has the same answer, but falls in QIII, so it is

not correct)

Evaluating Inverse Functions

When looking for an inverse answer on the calculator, use the 2nd key first, then hit sin, cos, or tan.

When looking for an angle always hit the 2nd key first.

Last example: ◦ Degree mode, 2nd, sin, .5 = 30. ◦ Radian mode: 2nd, sin, .5 = .524 (which is pi/6)

Calculator

Find the value of: Sin-1 2

◦ This means at what angle is the sin = 2 ?◦ What does your calculator read? Why?

◦ 2 falls outside the domain of an inverse sine wave

Evaluating Inverse Functions

Cosine Wave

Domain and range of restricted wave? Domain and range of the inverse?

Cosine Wave

D: 0,R : 1,1

D: 1,1R : 0,

We must restrict the domain Now the inverse

Cosine Wave

Quadrant I is 0,2Quadrant I I is ,2

Tangent Wave

We must restrict the domain Now the inverse

Tangent Wave

Graphing Utility: Graph the following inverse functions.

a. y = arcsin x

b. y = arccos x

c. y = arctan x

–1.5 1.5

–

–1.5 1.5

2

–

–3 3

–

Set calculator to radian mode.

Graphing Utility: Approximate the value of each expression.

a. cos–1 0.75 b. arcsin 0.19

c. arctan 1.32 d. arcsin 2.5

Set calculator to radian mode.

Previously learned notation: ◦ fog(x) gof(x)

Composition of Functions

Find tan(arctan(-5))◦ -5 (the tangent and its inverse cancel each other

out!) Find arcsin(sin )

◦ The domain of the sine function is . Since is outside that domain, we’ll just say that the answer is: is outside the domain (unless you remember coterminal angles and can tell me the actual answer is

Find ◦ Outside the domain

Composition of Functions Using Inverse Properties

53

,2 2 5

3

53

3

cos(cos- )

Find the exact value of Draw the graph using only the info inside

the parentheses. (Easy way—completely ignore the fact that you have inverses!)

Composition of Functions2tan arccos3

Example: 2Find the exact value of tan arccos .3

x

y

3

2

cos , x 2, 3 xSince rr

2 23 2 5

y 5tan x 2

Positive so draw in Q1)

Find the

You Try1 3cos sin 5

45