CK-12-Calculus

6.5. Derivatives and Integrals Involving Inverse Trigonometric Functions http://www.ck12.org

6.5 Derivatives and Integrals Involving Inverse Trigonometric Functions

Learning Objectives

A student will be able to:

• Learn the basic properties inverse trigonometric functions.


• Learn how to use the derivative formula to use them to find derivatives of inverse trigonometric functions.


• Learn to solve certain integrals involving inverse trigonometric functions.

A Quick Algebraic Review of Inverse Trigonometric Functions

You already know what a trigonometric function is, but what is an inverse trigonometric function? If we ask what is
sin(π/ 6 )equal to, the answer is( 1 / 2 ).That is simple enough. But what if we ask what angle has a sine of( 1 / 2 )?
That is an inverse trigonometric function. So we say sin(π/ 6 ) = ( 1 / 2 ),but sin−^1 ( 1 / 2 ) = (π/ 6 ).The “sin−^1 ” is the
notation for the inverse of the sine function. For every one of the six trigonometric functions there is an associated
inverse function. They are denoted by

sin−^1 x,cos−^1 x,tan−^1 x,sec−^1 x,csc−^1 x,cot−^1 x

Alternatively, you may see the following notations for the above inverses, respectively,

arcsinx,arccosx,arctanx,arcsecx,arccscx,arccotx

Since all trigonometric functions are periodic functions, they do not pass the horizontal line test. Therefore they are
not one-to-one functions. The table below provides a brief summary of their definitions and basic properties. We
will restrict our study to the first four functions; the remaining two, csc−^1 and cot−^1 ,are of lesser importance (in
most applications) and will be left for the exercises.

TABLE6.1:

Inverse Function Domain Range Basic Properties

sin−^1 − 1 ≤x≤ 1 − 2 π≤y≤π 2 sin(sin−^1 (x)) =x

cos−^1 − 1 ≤x≤ 1 0 ≤y≤π cos−^1 (cosx) =

cos(cos−^1 (x)) =x

tan−^1 allR (− 2 π,π 2 ) tan−^1 (tanx) =

tan(tan−^1 (x)) =x

sec−^1 (−∞,− 1 ]∪[ 1 ,+∞) [ 0 ,π 2 )∪(π 2 ,π] sec−^1 (secx) =

sec(sec−^1 (x)) =x