Chapter 4, Problem 33RE

To determine

To find: the exact value of the six trigonometric functions of the angle $\theta$ .

Answer to Problem 33RE

  $\begin{array}{l}\sin \theta =\frac{4}{\sqrt{41}},\cos \theta =\frac{5}{\sqrt{41}},\tan \theta =\frac{4}{5}\\ \cot \theta =\frac{5}{4},\csc \theta =\frac{\sqrt{41}}{4},\sec \theta =\frac{\sqrt{41}}{5}\end{array}$

Explanation of Solution

Given information:

Consider the below right angle triangle

Formula used:

Consider the given expression $\cot \left(\arcsin x\right)=\frac{\sqrt{1-x^2}}{x}$

Calculation:

Consider the given right triangle $ABC$

Apply Pythagorean theorem.

  $\begin{array}{l}A{C}^2=A{B}^2+B{C}^2\\ A{C}^2=5^2+4^2\\ AC=\sqrt{25+16}\\ AC=\sqrt{41}\end{array}$

Find the trigonometric functions

  $\begin{array}{l}\sin \theta =\frac{BC}{AC}=\frac{4}{\sqrt{41}}\\ \cos \theta =\frac{AB}{AC}=\frac{5}{\sqrt{41}}\\ \tan \theta =\frac{BC}{AB}=\frac{4}{5}\\ \cot \theta =\frac{AB}{BC}=\frac{5}{4}\\ \csc \theta =\frac{AC}{AB}=\frac{\sqrt{41}}{4}\\ \sec \theta =\frac{AC}{BC}=\frac{\sqrt{41}}{5}\end{array}$

Hence, the exact values of six trigonometric functions are determined.