Chapter 13.2, Problem 4OE

To determine

To find: The exact values of the trigonometric functions of the number $s$ .

Answer to Problem 4OE

The values of trigonometric functions are

  $\sin s=\frac{4}{5},\cos s=\frac{3}{5},\tan s=\frac{4}{3},\cot s=\frac{3}{4},\sec s=\frac{5}{3},\cos ecs=\frac{5}{4}$ .

Explanation of Solution

Given information:

Each circle is a unit circle centered at the origin.

  

Calculation:

This is a unit circle, the angle is ( in radians ) that the line from the origin to the point with the $x-axis$ is equal to the arc length $s$

Therefore, $\sin s$ is the $y-$ co-ordinate of the given point.

  $\sin s=\frac{4}{5}$

And $\cos s$ is the $y-$ co-ordinate of the given point.

  $\cos s=\frac{3}{5}$

Calculate the other four values.

  $\tan s$

  $\begin{array}{c}\tan s=\frac{\sin s}{\cos s}\\ \tan s=\frac{\raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$5$}\right.}{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$5$}\right.}\\ =\frac{4}{3}\end{array}$

  $\cot s$

  $\begin{array}{l}\cot s=\frac{\cos s}{\sin s}\\ \cot s=\frac{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$5$}\right.}{\raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$5$}\right.}\\ \cot s=\frac{3}{4}\end{array}$

  $\sec s$

  $\begin{array}{l}\sec s=\frac{1}{\cos s}\\ \sec s=\frac{1}{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$5$}\right.}\\ \sec s=\frac{5}{3}\end{array}$

  $\cos ecs$

  $\begin{array}{l}\cos ecs=\frac{1}{\sin s}\\ \cos ecs=\frac{1}{\raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$5$}\right.}\\ \cos ecs=\frac{5}{4}\end{array}$

Therefore, the values of trigonometric functions are $\sin s=\frac{4}{5},\cos s=\frac{3}{5},\tan s=\frac{4}{3},\cot s=\frac{3}{4},\sec s=\frac{5}{3},\cos ecs=\frac{5}{4}$ .