Page 202 - Calculus Demystified

Transcendental Functions
CHAPTER 6
6.6.5 OTHER INVERSE TRIGONOMETRIC FUNCTIONS 189
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,π)
(Fig. 6.21). Then Cot is decreasing on that interval and takes on all real values.
Therefore the inverse
Cot −1 : (−∞, ∞) → (0,π)

y

y = Cot x

x

Fig. 6.21

is well defined. Look at Fig. 6.22 for the graph. It can be shown that
d −1 1
Cot x =− .
dx 1 + x 2

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 recip-
rocal, 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]