OBJECTIVES:

Evaluate the inverse trigonometric functions

Evaluate the compositions of trigonometric functions

Inverse Trigonometric Functions

RECALL: for a function to have an inverse function, it must be one-to-one – that is, it must pass the Horizontal Line Test.

So consider the graphs of the six trigonometric functions, will they pass the Horizontal Line Test?

Inverse functions

However, if you restrict the domain of the trig functions, you will have a unique inverse function. But in such a restriction, the range will be unchanged, it will take on the full range of values for the trig function. Therefore, allowing the trig function to be one-to-one.

The INVERSE SINE FUNCTION is defined by

where the domain is and the range is

Inverse Trigonometric functions

1arcsin or sin iff siny x y x y x 1,1 ,

2 2

EX 1: If possible, find the exact valueA)

C)

B)

D)

1arcsin

21 3

sin2

1sin 1 arcsin 2

The INVERSE COSINE FUNCTION is defined by

where the domain is and the range is

The INVERSE TANGENT FUNCTION is defined by

where the domain is and the range is

Inverse Trigonometric functions

1arccos or cos iff cosy x y x y x 1,1 0,

1arctan or tan iff tany x y x y x , ,

2 2

EX 2: If possible, find the exact value A)

C)

B)

D)

arccos1 1 2cos

2

1tan 1 arctan 0

Function Domain Range Quadrant of the Unit

Circle Range Values come

from

I and IV

I and II

I and IV

I and II

I and II

I and IV

Inverse Trigonometric Functions

arcsiny x

arccosy x

arctany x

arccoty x

arcsecy x

arccscy x

1,1

1,1

,

,

, 1 1,

, 1 1,

,2 2

0,

,2 2

0,

0, ,2

y

, , 02 2

y

A)

B)

C)

EX 3: Use a calculator to approximate the value, if possible

cot( 0.3541)arc

arcsin 0.92837781

sec 1.2871684arc

A)

EX4: Find the exact value of the composition function

3sin arctan

2

B)

EX4: Find the exact value of the composition function

5tan arccos

13

C)

EX4: Find the exact value of the composition function

cos cos 0.5arc

D)

EX4: Find the exact value of the composition function

5arccos cos

4