Inverse Trigonometric Functions and Their Derivatives · Inverse Trigonometric Functions and Their...

Inverse Trigonometric Functions and Their Derivatives

None of the trigonometric functions satisfies the horizontal line test, so none of them has an inverse.

The inverse trigonometric functions are defined to be the inverses of particular parts of the trigonometric functions; parts that do have inverses.

y = sin(x)

2 2x

π π− ≤ ≤

sin −1(x) = arcsin(x)

y = cos(x)0 x π≤ ≤

cos−1(x) = arccos(x)

y = tan(x)

tan−1(x) = arctan(x)

2 2x

π π− ≤ ≤

y = sec(x)

sec−1(x) = arcsec(x)

0

2

x

x

ππ

≤ ≤

≠

Evaluating inverse trigonometric functions

The equation is equivalent with the equation

Thus sin −1(x) should be thought of as the angle whose sine is x. However, many angles have sine equal to x, and in this case, we want the angle that lies between and

. A similar idea holds for all the other inverse

trigonometric functions. Each is an angle, but you must choose the particular angle that satisfies the restriction appropriate to that function.

2π

2π−

1sin ( )y x−=

sin( ).x y=

Example. Find sin−1(−1) and cos−1(1/2)

Solution. 1. Let y = sin−1(−1). Then sin(y) = −1, so

2y π=−

2. Let y = cos−1(1/2). Then cos(y) = 1/2, so 3

y π=

We can also find identities involving the inverse trigonometric functions, by using ordinary trigonometric identities.

Example. Show that cos(sin−1(x)) = 21 x−

1sin ( )x−x

1

21 x−

The result is obvious from the diagram:

Example. Clearly sin(sin−1(x)) = x. Is it also true thatsin−1(sin (x)) = x?

Actually, this is not true because of the restrictions on the inverse sine. The graph of sin−1(sin (x)) is shown below.

For example, if x = then sin(x) = . But the angle in

the range having that sine is not .

54π 1

2

2 2xπ π− ≤ ≤ 5

4π

4π

All such expressions (inversetrigfn(trigfn) = ) must be treated with care.

Derivatives of Inverse Trigonometric Functions

Theorem. 11sin ( )21

d xdx x

− = −

Proof. 1sin ( )y x−=Let so sin( ) . Theny x=

cos( ) 1dyydx

=

1 1 1cos( ) 2 21 sin ( ) 1

dydx y y x

= = =− −

In a similar way we can show that

11cos ( )21

d xdx x

−− = −

Theorem. 11tan ( )21

d xdx x

− = +

Proof. 1tan ( )y x−=Let so tan( ) . Theny x=

2sec ( ) 1dyydx

=

1 1 12 2 2sec ( ) 1 tan ( ) 1

dydx y y x

= = =+ +

Theorem. 11sec ( )2| | 1

d xdx x x

− = −

Proof. 1sec ( )y x−=Let so sec( ) . Theny x=

sec( )tan( ) 1dyy ydx

=

1 cos( )cos( )sec( ) tan( ) sin( )

dy yydx y y y

= =

y

1

x 2 1x −

2cos ( ) 1 12sin( ) 2 21 | | 1

y xy x x x x

= = =− −

We have now shown the following rules for differentiation.

Basic Rule Generalized Rule

111. sin ( )21

d xdx x

− = −

11sin ( )21

d duudx dxu

− = −

112. tan ( )21

d xdx x

− = +

11tan ( ) 21

d duudx dxu

− = +

113. sec ( )2| | 1

d xdx x x

− = −

11sec ( )2| | 1

d duudx dxu u

− = −

The other three inverse trig functions have derivatives that are the negatives of their respective cofunctions.

Problem. Suppose that . Find the exact values

of

11cos2

θ −=

sin( ),tan( ),cot( ),sec( ),csc( ).θ θ θ θ θ

Solution. Construct the following triangle.

θ1

23 so that cos( ) = θ 1

2

Miscellaneous Problems

θ1

23

The other functions can be read directly from that triangle.

3sin( )2

θ =

tan( ) 3θ = 1cot( )3

θ =

sec( ) 2θ = 2csc( )3

θ =

Problem. Compute . 31sec sin4

− −

θ

43−

Solution. Construct the following triangle.

7

Then sec(θ) = 47

Problem. Complete the identity . 1tan cos ( ) ?x− =

x

Solution. Construct the following triangle.

Then tan(θ) = θ

121 x−

21 xx−

Problem. Find the derivative of . 1cos (2 1)x− +

Solution.

1 21cos (2 1) (2 1)2 21 (2 1) 1 (2 1)

d dx xdx dxx x

− + =− + =− − + − +

Problem. Find the derivative of . 1tan ( )x−

Solution.

1 1 1 11tan ( ) ( )2 (1 )2 2 (1 )(1 ( ) )

d dx xdx dx x x x xx

− = = = + + +

Problem. Find the derivative of . 1sec ( )xe−

Solution.

( )1 1 11sec ( )

2 2 21 1| | 1

d dx x xe e edx dx x x xe e ex xe e

− = = = − −−

Problem. Find the derivative of . 1ln sin ( )x−

Solution.

( ) 1 1 11 1ln sin ( ) sin ( )1 1 2sin ( ) sin ( ) 1

d dx xdx dxx x x

− − = = − − −

Problem. Find the derivative of . 1tan ( )x−

Solution.

1 1 11 1tan ( ) tan ( )21 1 12 tan ( ) 2 tan ( )

d dx xdx dx xx x

− − = = − − +

Problem. Find the derivative of . 1 1sin ( ) cos ( )x x− −+

Solution. 1 11 1sin ( ) cos ( ) 02 21 1

d x xdx x x

−− − + = + = − −

Why? Because 1 1sin ( ) cos ( )2

x x π− −+ =

Problem. Find the derivative of . 312 sin ( )x x−

Solution.

3 21 2 1 12 sin ( ) 3 sin ( ) sin ( )dx x x x xdx

− − − = +

( )32 1sin ( )d x xdx

−

3 2 11 2 12 sin ( ) 3 sin ( )21

x x x xx

− − = + −

( )21sin ( )31 22 sin ( ) 321

xx x x

x

−− = +

−

Problem. Find the derivative of . ( )1sin tan ( )x−

Solution.

11cos tan ( )21

xx

− = +

( )1sin tan ( )d xdx

−

1 1cos tan ( ) tan ( )dx xdx

− − =

Problem. Find by implicit differentiation, if . 3 1tan ( ) yx x y e−+ =

Solution.

dydx

2 13 tan ( )21

x dy dyyx y edx dxy

−+ + =+

so 2 13 tan ( )21

x dy dyye x ydx dxy

−− =− −+

2 13 tan ( )21

x y dye x ydxy

−− =− − +

2 13 tan ( )

21

dy x ydx x ye

y

−− −=

− +