The Inverse of Trigonometric Functions

Find the Exact Value of all Trigonometric Functions

Find the Approximate Value of the Inverse of Trigonometric Functions

Review

Chapter 4We discussed inverse functions and we noted

that if a function is one to one it will have an inverse function

We also discussed that if a function is not one to one it may be possible to restrict the domain in some manner so that the restricted function is one to one

Properties of a one to one function

1. f-1(f(x))=x for every x in the domain of f and f(f-1(x)) =x for every x in the domain of f-1

2. Domain of f = range of f-1, and the range of f = domain of f-1

3. The graph of f and graph of f-1 are symmetric with respect to the line y=x

4. If a function y=f(x) has an inverse function, the equation of the inverse function is x= f(y). The solution of this equation is y=f-1(x)

The inverse sine function

Graph of the functionBecause every horizontal line y=b where b is

between -1 and intersects the graph of y=sin xInfinitely many times it follows from the horizontal

line test that the function is not one to one.However if we restrict the domain of the function

between the restricted function is one to one and hence will have an inverse function

2,

2

How do we find the inverse

An equation fro the inverse of y=f(x) = sin x is obtained by changing x and y.

The explicit form is called the inverse sine of x and is denoted by y = f-1(x) and sin-1 x

y= sin-1 x means x = sin yWhere -1≤ x ≤ 1 and

22

y

Finding the exact Value of an Inverse sine function

Find the exact value of: sin-1 1

Find the exact value of: sin-1

2

1

Finding the Approximate Value of an Inverse Sine

Find the approximate value of sin-1

Round your answer to the nearest hundredth of a radian

Find the approximate value of sin-1

Round your answer to the nearest hundredth of a radian

3

1

4

1

Showing they are inverses

f-1(f(x))= sin-1(sin x) = x, where f(f-1(x)) = sin (sin-1 x) = x where -1≤ x≤ 1

22

x

Finding the exact value of a compositefunction of sine

8sinsin 1

Inverse Cosine

Graph cos x

We are going to have to restrict the domain of the cosine also but differently from the sine function. Why?

Inverse cosine

We need to interchange our x and y to find the inverse of the cosine

y = cos-1 x means x= cos ywhere -1≤x≤1 and 0≤ y ≤ π

Finding the exact value

Find the exact value of: cos-1 0

Find the exact value of: cos-1 2

2

Definition of inverse

f-1(f(x)) = cos-1(cos x) = x where 0≤x≤π

f(f-1(x)) = cos(cos-1 x) = x where -1≤x ≤ 1

Finding the exact value of a composite function

Find the exact value of :

12coscos 1

The inverse tangent function

Again for the tangent function we must restrict the domain of y=tan x to the interval of

y = tan-1x means x = tan yWhere -∞<x<∞ and

22

x

22

y

Finding the exact value of an inverse tangent function

Find the exact value of: tan-1 = 1

Find the exact value of: tan-1( )3

The remaining Inverse trigonometric functions

y = sec-1x means x= sec ywhere |x|≥ 1 and 0≤ y ≤ π, y≠

y = csc-1x means x=csc y

where |x|≥ 1 and

y= cot-1 x means x= cot ywhere -∞<x<∞ and 0<y<π

2

0,22

yy

Classwork

Page 423 examples 13-44 every fourth problemPage 429 examples 9-41 first column

Homework

Page 429 10-54 second column