Inverse Trig Functions and Differentiation By Dr. Julia Arnold.

Inverse Trig Functions and Differentiation

ByDr. Julia Arnold

Since a function must pass the horizontal line test to have an inverse function, the trig functions, being periodic, have to have their domains restricted in order to pass the horizontal line test.For example: Let’s look at the graph of sin xon [-2pi, 2 pi].

-6.0-5.0-4.0-3.0-2.0-1.0 1.02.03.04.05.06.07.0

-1.0

1.0

x

y

Flunks thehorizontalline test.

By restricting the domain from [- pi/2, pi/2] we produce a portion of the sine function which will pass the horizontalline test and go from [-1,1].

The inverse sine function is written as y = arcsin(x)which means that sin(y)=x.Thus y is an angle and x is a number.

-1.0 1.0

-1.0

1.0

x

y-1.0 1.0

-1.0

1.0

x

y

Y = sinx

Y = arcsin x

We must now do this for each of the other trig functions:

1.0 2.0 3.0x

Y = cos x on [0, pi]Range [-1,1]

-1.0 1.0 2.0

-2.0

-1.0

1.0

2.0

3.0

x

y

Y= arccos (x) on [-1,1]range [0, pi]

arctan(x) has a range of (- pi/2, pi/2)

arccot(x) has a range of (0, pi)

arcsec(x) has a range of [0, pi], y pi/2

arccsc(x) has a range of [- pi/2, pi/2], y 0

Evaluate without a calculator:

2

1arcsin

x2

1arcsin

Step 1: Set equal to x

Step 2: Rewrite as 2

1sin

x

Step 3: Since the inverse is only defined in quadrants 1 & 4 for sin we are looking for an angle in the 4th quadrant whose value is -1/2.

The value must be - pi/6.


0arccos

x0arccosStep 1: Set equal to x

Step 2: Rewrite as 0cos x

Step 3: The inverse is only defined in quadrants 1 & 2 for cos so we are looking for the angle whose value is 0.

The value must be pi/2.


3arctan

x3arctan

3tan x

Step 3: The inverse is only defined in quadrants 1 & 4 for tan so we are looking for the angle whose tan value is sqr(3).

The tan 60 = sqr(3) or x = pi/3

Evaluate with a calculator:

3.arcsin

x3.arcsin

Check the mode setting on your calculator. Radianshould be highlighted.Press 2nd function sin .3 ) Enter.The answer is .3046

Inverse Properties

If -1 < x < 1 and - pi/2 < y < pi/2 then sin(arcsin x)=x and arcsin(siny)=y

If - pi/2 < y < pi/2 then tan(arctan x)=x and arctan(tany)=y

If -1 < x < 1 and 0 < y < pi/2 or pi/2 < y < pi then sec(arcsec x)=x and arcsec(secy)=y

On the next slide we will see how these properties are applied

Inverse Properties Examples

If - pi/2 < y < pi/2 then tan(arctan x)=x and arctan(tany)=y

Solve for x:

Thus: tan arctan(2 3) tan

4x

132 x

2

42

x

x

arctan(2 3)4

x


If where 0 < y < pi/2 find cos y

Solution: For this problem we use the right triangle

Sin(y) = x, thus the opp side must be xand the hyp must be 1, so sin y = x

x 1

By the pythagorean theorem, this makesthe bottom side

21 x21 x

Cos(y) = 21 x

arcsin( )y x


If find tan y

Solution: For this problem we use the right triangle

1

By the pythagorean theorem, this makesthe opp side

114525 22

tan(y) =

2

5

2

1

5sec

2y arc

Derivatives of the Inverse Trig Functions

21

)arcsin(

u

u

dx

ud

21

)arccos(

u

u

dx

ud

21

)arctan(

u

u

dx

ud

21

)cot(

u

u

dx

uarcd

1

)sec(2

uu

u

dx

uarcd 1

)csc(2

uu

u

dx

uarcd

Examples Using the Derivatives of the Inverse Trig Functions

21

)arcsin(

u

u

dx

ud

Find the derivative ofarcsin 2x

Let u = 2x du/dx = 2

Note: u’ = du/dx

2 2

arcsin(2 ) 2 2

1 41 2

d x

dx xx


21

)arctan(

u

u

dx

ud

Find the derivative ofarctan 3x

Let u = 3x du/dx = 3

Note: u’ = du/dx

2

arctan(3 ) 3

1 9

d x

dx x


Find the derivative of

Letx

x

edx

du

eu

2

2

2

1

)sec(2

uu

u

dx

uarcd

1

2

1

2)sec(4222

2

xxx

x

eee

e

dx

uarcd

2sec( )xarc e

Some more examples:

3sec arctan sec( )

5x

Solution: Use the right triangle on the coordinate graph

Let

5

3tan

5

3arctan

x

x

Now using the triangle we can find sec x after we find the hyp.

3425953 22 34

5

34)sec( x

find

Some more examples:

Write the expression in algebraic form x3arctansec

Solution: Use the right triangle

xy

yx

3tan

3arctan

Now using the triangle we can find the hyp.

Letthen

y

3x

1

222 9131 xx 291 x

22

911

91sec x

xy

Some more examples:

Find the derivative of: 21

arctanarctan)( xxxf

Let u = 21

x

xdx

du

2

1

xxx

xxf

12

1

1

2

1

)( 2

Please let us know if this presentation has been beneficial.

Thanks.

Inverse Trig Functions and Differentiation By Dr. Julia Arnold.

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