5.6 Inverse Trigonometric 5.6 Inverse Trigonometric Functions: DifferentiationFunctions: DifferentiationDevelop properties of the six inverse trig functionsDifferentiate an inverse trig functionReview basic differentiation rules for elementary functions.

None of the six basic trigonometric functions have an inverse. Surprised? Since they are periodic, they do not have an inverse unless you restrict the domain.

1

1

sec

csc

y x

y x

1

1

tan

cot

y x

y x

1

1

sin

cos

y x

y x

Notice each makes 1st quadrant part of its graph.

What is the purpose of inverse trig functions?

Ex 1 p372 Evaluating Inverse Trig Functions

1 1. sin

2a

. arccos0b

tan 3y arc

. arcsin 0.6d

By definition, y=arcsin ½ means sin y = ½ In the interval [-π/2, π/2], the angle with that sine value would be π/6

By definition, y=arccos 0 means cos y = 0 In the interval [0, π], the angle with that cosine value would be π/2

By definition, means

in the interval [-π/2, π/2], the angle with that tangent value would be -π/3

. arctan 3c tan 3y

In radian mode, type in sin-1 0.6 and it produces approximately 0.6435. If you multiply by 180 and divide by pi, you could get it in degrees, or just set to degree mode. Most often you will want radians.

1 1Inverse functions have the property that ( ( )) and ( ( ))f f x x f f x x

Remember that when applying these properties they have restricted domains. For x-values outside these domains the properties do not hold. For example arcsin(sin π)) = 0, not π.

Ex 2 p.373 Solving an equation

arctan (2x – 5)=π/4

tan(arctan (2x – 5))= tan π/4

2x – 5 = 1

x = 3

Sometimes you will have to evaluate expressions like cos(arcsin x)

Ex 3 p. 373 Using right triangles

. Given arccos , where 0 , find sin y2a y x y

yx = cos y

x

1

22x 1 21 x

Therefore sin y= 21 x

5. Given arcsec , find tan y

2b y

5sec

2y

5

2

y 2222 5 1

1tan y=

2

yx

1

Looking at derivatives for inverse trig functions, even though they are transcendental, their derivatives are algebraic.

arcsiny xsin y x cos ' 1y y

1'

cosy

y

222x 1

21 x

2

1'

1y

x

Ex 4 p. 374 Differentiating Inverse Trig Functions

d. arcsin 5

dxa x

2

5

1 25x

1d. tan 2

dxb x 2

2

1 4x

1d. cos

dxc x

1

2 1x x

2

1

2 x x

3d. sec

dxxd arc e

3

3 3 2

3

( ) 1

x

x x

e

e e

6

3

1xe

Ex 5 p. 374 A Derivative that can be simplified

Differentiate 2arcsin 1y x x x

12 22

2

1 1' (1 ) ( 2 ) 1 (1)

21y x x x x

x

22

2 2

1 ( )' 1

1 1

xy x

x x

22

2

1' 1

1

xy x

x

2 2' 1 1y x x

2' 2 1y x 2 21

1 arcsin 12

x dx x x x C

You can see one of the benefits of inverse trig functions – they can be used to integrate common algebraic functions.

Ex 6 p. 375 Analyzing an Inverse Trig Graph

Analyze the graph of y = (arctan x)2

Solution: From the derivative,

The only critical number occurs when arctan x = 0, which happens at x = 0.By the first derivative test, this turns out to be a minimum.

From the second derivative,

2

1' 2(arctan )

1y x

x 2

2arctan

1

x

x

22

22

2(1 ) 2(arctan )(2 )

1''

1

x x xx

yx

22

2(1 2 arctan )

1

x x

x

Finally looking at there are horizontal

asymptotes at y =

2

2lim arctan

4xx

2

4

3

2

1

-2 2

y=2

4

inflection points

The points of inflection occur when 2xarctan x = 1, and from Newton’s method, this occurs at x ≈ 0.765

Here are the rules you should have memorized by test on Jan 27

The number pi is mighty fine 3.14159The number e is very great 2.71828

Ted Mueller 2011

5.6 p. 377/ 1-65 EOO, 71-72, 94