Q. 122 - Precalculus Enhanced with Graphing Utilities

Question

Show that the range of the cotangent function is the set of all

real numbers.

Step-by-Step Solution

Answer

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

Step 1. Given Information

A co-tangent function.

Step 2. Explanation

$$\begin{gathered} Consider \\ cot=\frac{x}{y} \\ a=\frac{x}{y} \\ x=ay \\ But, x^2+y^2=1 \\ {a}^2{y}^2+y^2=1 \\ y=\pm \frac{1}{\sqrt{a^2+1}} \\ therefore the value of x is \\ x=ay \\ =a\times \pm \frac{1}{\sqrt{a^2+1}} \\ =\pm \frac{a}{\sqrt{a^2+1}} \\ thusfor sny real number a, there is a point p(x,y) for which cot t=a \\ i.e. -\infty <cot t<\infty thus range of co\tan get function is set of all real numbers. \end{gathered}$$