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Inverse Trigonometric Functions Help (page 2)

By — McGraw-Hill Professional
Updated on Sep 12, 2011

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

Transcendental Functions 6.6 Inverse Trigonometric Functions

we see that Tan x is increasing, hence it is one-to-one (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 one-to-one 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

Transcendental Functions 6.6 Inverse Trigonometric Functions

 

Transcendental Functions 6.6 Inverse Trigonometric Functions

Fig. 6.19

Transcendental Functions 6.6 Inverse Trigonometric Functions

Fig. 6.20

Solution 1

We have

Transcendental Functions 6.6 Inverse Trigonometric Functions

As with the first two trigonometric functions, we note that the tangent function takes each of the values Transcendental Functions 6.6 Inverse Trigonometric Functions 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

Transcendental Functions 6.6 Inverse Trigonometric Functions

Next we calculate some derivatives:

Example 2

Calculate the following derivatives:

Transcendental Functions 6.6 Inverse Trigonometric Functions

Solution 2

We have

Transcendental Functions 6.6 Inverse Trigonometric Functions

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

Transcendental Functions 6.6 Inverse Trigonometric Functions

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

Transcendental Functions 6.6 Inverse Trigonometric Functions

Solution 1

For clarity we set φ( x ) = cos x, φ′ ( x ) = − sin x . The integral becomes

Transcendental Functions 6.6 Inverse Trigonometric Functions

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

Transcendental Functions 6.6 Inverse Trigonometric Functions

You Try It : Calculate ∫ x /(1 + x 4 ) dx .

Example 2

Calculate the integral

Transcendental Functions 6.6 Inverse Trigonometric Functions

Solution 2

For clarity we set φ( x ) = x 3 , φ′( x ) = 3 x 2 . The integral then becomes

Transcendental Functions 6.6 Inverse Trigonometric Functions

We know that this last integral equals

Sin −1 φ ( x ) + C .

Resubstituting the formula for φ gives a final answer of

Transcendental Functions 6.6 Inverse Trigonometric Functions

You Try It : Evaluate the integral

Transcendental Functions 6.6 Inverse Trigonometric Functions

Find practice problems and solutions for these concepts at: Transcendental Functions Practice Test.

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