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Inverse Trig Functions and Differentiation

- By
- Dr. 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 Lets look at

the graph of sin x on -2pi, 2 pi.

Flunks the horizontal line test.

By restricting the domain from - pi/2, pi/2 we

produce a portion of the sine function which will

pass the horizontal line 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.

Y sinx

Y arcsin x

We must now do this for each of the other trig

functions

Y cos x on 0, pi Range -1,1

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

Step 1 Set equal to x

Step 2 Rewrite as

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.

Evaluate without a calculator

Step 1 Set equal to x

Step 2 Rewrite as

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.

Evaluate without a calculator

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

Check the mode setting on your calculator.

Radian should be highlighted. Press 2nd function

sin .3 ) Enter. The answer is .3046

Inverse Properties

If -1 lt x lt 1 and - pi/2 lt y lt pi/2 then

sin(arcsin x)x and arcsin(siny)y

If - pi/2 lt y lt pi/2 then tan(arctan x)x and

arctan(tany)y

If -1 lt x lt 1 and 0 lt y lt pi/2 or pi/2 lt y lt pi

then sec(arcsec x)x and arcsec(secy)y

On the nextwe will see how these

properties are applied

Inverse Properties Examples

Solve for x

If - pi/2 lt y lt pi/2 then tan(arctan x)x and

arctan(tany)y

Thus

Inverse Properties Examples

If

where 0 lt y lt pi/2 find cos y

Solution For this problem we use the right

triangle

Sin(y) x, thus the opp side must be x and the

hyp must be 1, so sin y x

1

x

By the pythagorean theorem, this makes the bottom

side

Cos(y)

Inverse Properties Examples

find tan y

If

Solution For this problem we use the right

triangle

By the pythagorean theorem, this makes the opp

side

1

2

tan(y)

Derivatives of the Inverse Trig Functions

Examples Using the Derivatives of the Inverse

Trig Functions

Note u du/dx

Find the derivative of arcsin 2x

Let u 2x du/dx 2

Examples Using the Derivatives of the Inverse

Trig Functions

Note u du/dx

Find the derivative of arctan 3x

Let u 3x du/dx 3

Examples Using the Derivatives of the Inverse

Trig Functions

Find the derivative of

Let

Some more examples

Let

find

Solution Use the right triangle on the

coordinate graph

Now using the triangle we can find sec x after

we find the hyp.

Some more examples

Write the expression in algebraic form

Let then

Solution Use the right triangle

Now using the triangle we can find the hyp.

3x

y

1

Some more examples

Find the derivative of

Let u

Please let us know if this presentation has been

beneficial. Thanks.

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