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Precalculus Enhanced with Graphing Utilities

Sullivan

$Math Studyset 91影视 Explanations$ Math

6th Edition 路 1200 pages

ISBN: 9780321795465

Chapter 6: Q. 122 (page 397)

Show that the range of the cotangent function is the set of all
real numbers.

Short Answer

Expert verified

The range of cotangent function is a set of real numbers.

Step by step solution

01

Step 1. Given Information

A co-tangent function.

02

Step 2. Explanation

$C o n s i d e r c o t = \frac{x}{y} a = \frac{x}{y} x = a y B u t, x^{2} + y^{2} = 1 a^{2} y^{2} + y^{2} = 1 y = 卤 \frac{1}{\sqrt{a^{2} + 1}} t h e r e f o r e t h e v a l u e o f x i s x = a y = a 脳卤 \frac{1}{\sqrt{a^{2} + 1}} = 卤 \frac{a}{\sqrt{a^{2} + 1}} t h u s f o r s n y r e a l n u m b e r a, t h e r e i s a p o i n t p (x, y) f o r w h i c h c o t t = a i . e . - 鈭� < c o t t < 鈭� t h u s r a n g e o f c o \tan g e t f u n c t i o n i s s e t o f a l l r e a l n u m b e r s .$

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