8.2: Inverse Trig Functions

© 2008 Roy L. Gover(www.mrgover.com)

Learning Goals:

•Review Special Angles

•Evaluate inverse trig functions.

ReviewThe special angles are:

6

3

60° or

4

45° or

30° or

Consider the first two special angles in degrees...

30°

60°

Long Side

Short

Sid

e

Hypotenus

e

Review

And in radians:

Long Side

Short

Sid

e

Hypotenus

e

Review

6

3

Important IdeaIn a 30° ,60° ,

triangle:•the short side is one-

half the hypotenuse.

•the long side is times the short side.

3

( 6) ( 3)( 2)90°

Important IdeaIn a 45° ,45° ,90° triangle

:•The legs of the triangle are equal.

•the hypotenuse is times the length of the leg.

2

( 4) ( 4)( 2)

Try ThisFind the length of the missing sides: 30

°

60°1

2

1 3 or 3

Try ThisFind the length of the missing sides

1

12

45°

Important IdeaMany trig functions can be solved without graphing by using special angles and inverse trig functions. A special angle solution will be an exact solution whereas a graphing solution is only approximate.

DefinitionTrig Function

Inverse Trig Function

siny x 1sin x y

cosy x 1cos x y

tany x 1tanx y

Important Idea

arcsin y

arccos y

arctan y

In some books:1sin y

1cos y

1tan y

instead ofinstead ofinstead of

ExampleFind the exact value without using a calculator;

1 1sin

2 1 2

cos2

1sin 1

1tan (1) 1 3cos

2

Example

Find all values of x in the interval for which:

2cos

2x

0 360x

Example

Find all values of x in the interval for which:

2cos

2x

0 2x

Try ThisFind all values of x in the interval for which: 1

sin2

x

0 360x

210 & 330x

Try ThisFind all values of x in the interval for whichtan 1x

0 360x

45 & 225x

Hint: in what quadrants is the tangent positive?

Example

Write each equation in the form of an inverse relation:

4tan

5

Example

Write each equation in the form of an inverse relation:

1cos

3

Try This

Write each equation in the form of an inverse relation:sin 1x

1sin 1 or arcsin1x x

ExampleFind the value of x in the for which:

cos .6328x

Leave your answer in degrees to the nearest tenth.

Try ThisFind the value of x in the for which:sin .6328x Leave your answer in degrees to the nearest tenth.

x=39.3

Can you find another value for x?

DefinitionThe calculator will provide only the Principal Values of inverse trig functions:

1sin x

1cos x

1tan ( )x 90 ,90 2, 2

0,180 0,or

or

ExampleEvaluate the expression. Assume all angles are in quadrant 1.

1sin arcsin

2

ExampleEvaluate the expression. Assume all angles are in quadrant 1.

3sin arccos

2

Try ThisEvaluate the expression. Assume all angles are in quadrant 1. 1

os arcsinc2

3

2

Racetrack curves are banked so that cars can make turns at high speeds. The proper banking angle,, is given by:2

tanv

gr

where v is the velocity of the car, g is the acceleration of gravity & r is the radius of the turn. Find when r=1000ft & v=180 mph.

Example

Lesson Close

In order to have an inverse trig function, we must restrict the domain so that duplicate values are eliminated.