4.7 Inverse Trigonometric Functions

*Intro to Inverse Functions*One-to-one

*Inverse of Sine

Inverse Functions

• f-1 (x) is the inverse of f(x)– If and only if the domain of f(x) is equal to the

range of f-1(x) AND the range of f(x) is equal to the domain of f-1(x)• In English the x and y’s switch

• Examplex y

1 -12

2 -13

3 -14

4 -15

x y

-12 1

-13 2

-14 3

-15 4

f(x)= f-1(x)=

Do all functions have inverses?

• Linear Functions • Quadratic Functions

• Cubic Functions • Absolute Value

One-to-One

• A function must be one-to-one to have an inverse

• One-to-one: there is only one x for each y– Horizontal Line Test– Solve for y

• You can write a restriction on the domain to make a function one-to-one

Trig Functions – Sine

Is y = sinx one-to-one??

Inverse of Sine

• f(x) = sinx has an inverse over the interval [- /2p , /2p ]

– sin x is increasing over the interval

– takes on the full range values -1 < y < 1

– sine is one-to-one

Inverse of Sine

• Inverse of sine is called:– inverse sine: sin-1

– arcsine: arcsin• the angle (or arc) whose sine is x

• Sine: input = angle, output = ratio of the sides• Arcsine: input = ratio of sides, output = angle

Common Mistake

• Don’t be FOOLED!!!

• sin-1 is not the same as 1/sin

Arcsine

• Definition: – The inverse sine function is defined by:y = arcsin iff siny = x

where -1 < x < 1 and –p/2 < y < p/2

• y = arcsin Domain: [-1,1]Range: [–p/2,

p/2]

Evaluating the Inverse Sine Function

• If possible, find the exact value.a) arcsin(-1/2)

b) sin-1(√3/2)

c) sin-1 2

Graph Arcsine by Hand

xy

y = sin x

y = arcsin x

xy

Cosine

Is cosine one-to-one??

Arccosine

• Definition: – The inverse cosine function is defined by:y = arccosx iff cosy = x

where -1 < x < 1 and 0 < y < p

• y = arccosx Domain: [-1,1]Range: [0,

p]

Graph of Arccosine

Evaluating Arccos

• arccos(1/√2)

• cos-1 (-1)

• arccos(0)

Tangent

Arctangent

• Definition: – The inverse tangent function is defined by:y = arctanx iff tany = x

where -∞ < x < ∞ and –p/2 < y < /2p

• y = arctanx Domain: (-∞,∞)Range: (-

p/2, /2p )

Graph of Arctan

Evaluating Inverse Tangent

• arctan 0

• tan-1 (-1)

• arctan(√3)

Evaluating on the Calculator

• Remember!!! Your input is the ratio of the sides and your output is the angle

• The mode is the units your answer (angle) will be in

• An error message is most likely because you are entering a number that is not in the domain

Evaluating Compositions of Functions

• tan(arccos2/3)

• cos(arcsin(-3/5))

• sin(arctan2)