**Introduction to Other Inverse Trigonometric Functions**

The most important inverse trigonometric functions are Sin ^{−1}, Cos ^{−1}, and Tan ^{−1}. We say just a few words about the other three.

Define Cot *x* to be the restriction of the cotangent function to the interval (0, ) (Fig. 6.21). Then Cot is decreasing on that interval and takes on all real values. Therefore the inverse

Cot ^{−1} : (−∞, ∞) → (0, )

is well defined. Look at Fig. 6.22 for the graph. It can be shown that

Define Sec *x* to be the function sec *x* restricted to the set [0, π/2) ∪ (π/2, π] (Fig. 6.23). Then Sec *x* is one-to-one. For these values of the variable *x* , the cosine function takes all values in the interval [−1, 1] except for 0. Passing to the reciprocal, we see that secant takes all values greater than or equal to 1 and all values less than or equal to −1. The inverse function is

Sec ^{−1} : (−∞, −1] ∪ [1, ∞) → [0, π/2) ∪ (π/2, π]

(Fig. 6.24). It can be shown that

The function Csc *x* is defined to be the restriction of Csc *x* to the set [− π/2, 0) ∪ (0, /2]. The graph is exhibited in Fig. 6.25. Then Csc *x* is one-to-one. For these values of the *x* variable, the sine function takes on all values in the interval [−1, 1] except for 0. Therefore Csc takes on all values greater than or equal to 1 and all values less than or equal to −1; Csc ^{−1} therefore has domain (−∞, −1] ∪ [1, ∞) and takes values in [−1, 0) ∪ (0, 1] (Fig. 6.26).

It is possible to show that

**You Try It** :

**Summary of Key Facts About the Inverse Trigonometric Functions**

**You Try It** : What is the derivative of Sec ^{−1} *x* ^{2} ?

**An Example Involving Inverse Trigonometric Functions**

**Example 1**

Hypatia is viewing a ten-foot-long tapestry that is hung lengthwise on a wall. The bottom end of the tapestry is two feet above her eye level. At what distance should she stand from the tapestry in order to obtain the most favorable view?

**Solution 1
**

For the purposes of this problem, view A is considered more favorable than view B if it provides a greater sweep for the eyes. In other words, form the triangle with vertices (i) the eye of the viewer, (ii) the top of the tapestry, and (iii) the bottom of the tapestry ( Fig. 6.27 ). Angle *α* is the angle at the eye of the viewer. We want the viewer to choose her position so that the angle *α* at the eye of the viewer is maximized.

The figure shows a mathematical model for the problem. The angle *α* is the angle *θ* less the angle *ψ* . Thus we have

*α = θ − ψ* = Cot ^{−1} ( *x* /12) − Cot ^{−1} ( *x* /2).

Notice that when the viewer is standing with her face against the wall then *θ = ψ = π* /2 so that *α* = 0. Also when the viewer is far from the tapestry then *θ − α* is quite small. So the maximum value for *α* will occur for some finite, positive value of *x* . That value can be found by differentiating *α* with respect to *x* , setting the derivative equal to zero, and solving for *x* .

We leave it to you to perform the calculation and discover that ft is the optimal distance at which the viewer should stand.

**You Try It** : Redo the last example if the tapestry is 20 feet high and the bottom of the tapestry is 6 inches above eye level.

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

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