The Inverse Tangent Function
Define the function Tan x to be the restriction of tan x to the interval (− /2, /2). Observe that the tangent function is undefined at the endpoints of this interval. Since
we see that Tan x is increasing, hence it is onetoone (Fig. 6.19). Also Tan takes arbitrarily large positive values when x is near to, but less than, /2. And Tan takes negative values that are arbitrarily large in absolute value when x is near to, but greater than, −π /2. Therefore Tan takes all real values. Since Tan : (− /2, /2) → (−∞, ∞) is onetoone and onto, the inverse function Tan ^{−1} : (−∞, ∞) → (− /2, /2) exists. The graph of this inverse function is shown in Fig. 6.20. It is obtained by the usual procedure of reflecting in the line y = x .
Example 1
Calculate
Solution 1
We have
As with the first two trigonometric functions, we note that the tangent function takes each of the values at many different points of its domain. But Tan x takes each of these values at just one point of its domain.
The derivative of our new function may be calculated in the usual way. The result is
Next we calculate some derivatives:
Example 2
Calculate the following derivatives:
Solution 2
We have
You Try It : Calculate ( d/dx ) Tan ^{−1} [ln x + x ^{3} ] and ( d/dx ) ln[Tan ^{−1} x ].
Integrals in Which Inverse Trigonometric Functions Arise
Our differentiation formulas for inverse trigonometric functions can be written in reverse, as antidifferentiation formulas. We have
The important lesson here is that, while the integrands involve only polynomials and roots, the antiderivatives involve inverse trigonometric functions.
Example 1
Evaluate the integral
Solution 1
For clarity we set φ( x ) = cos x, φ′ ( x ) = − sin x . The integral becomes
By what we have just learned about Tan ^{−1} , this last integral is equal to
−Tan ^{−1} φ( x ) + C .
Resubstituting φ( x ) = cos x yields that
You Try It : Calculate ∫ x /(1 + x ^{4} ) dx .
Example 2
Calculate the integral
Solution 2
For clarity we set φ( x ) = x ^{3} , φ′( x ) = 3 x ^{2} . The integral then becomes
We know that this last integral equals
Sin ^{−1} φ ( x ) + C .
Resubstituting the formula for φ gives a final answer of
You Try It : Evaluate the integral
Find practice problems and solutions for these concepts at: Transcendental Functions Practice Test.
 1

2
Ask a Question
Have questions about this article or topic? AskPopular Articles
 Kindergarten Sight Words List
 First Grade Sight Words List
 10 Fun Activities for Children with Autism
 Signs Your Child Might Have Asperger's Syndrome
 Theories of Learning
 A Teacher's Guide to Differentiating Instruction
 Child Development Theories
 Social Cognitive Theory
 Curriculum Definition
 Why is Play Important? Social and Emotional Development, Physical Development, Creative Development