Chapter 4, Problem 34RE

To determine

To Find:

The exact values of 6 trigonometric function of the angle $\theta$

Answer to Problem 34RE

Third side is $\sqrt{48}$ and

  $\begin{array}{l}\sin \theta =\frac{\sqrt{48}}{8}=0.866\\ \cos \theta =\frac{4}{8}=0.50\\ \tan \theta =\frac{\sqrt{48}}{4}=1.732\\ \cos ec\theta =\frac{8}{\sqrt{48}}=1.154\\ \sec \theta =\frac{8}{4}=2\\ \cot \theta =\frac{4}{\sqrt{48}}=0.577\end{array}$

Explanation of Solution

Given information:

Sides are given base b =4

Height h = 8

Formula used:

Pythagoras theorem formula.

  $h=\sqrt{p^2+b^2}$

And trigonometric formula

  $\sin \theta =\frac{perpendicular}{height}$

  $\cos \theta =\frac{base}{height}$

  $\tan \theta =\frac{perpendicular}{base}$

  $\cos ec\theta =\frac{1}{\sin \theta }$

  $\sec \theta =\frac{1}{\cos \theta }$

  $\cot \theta =\frac{1}{\tan \theta }$

Calculation:

the statement “ exact value of 6 trigonometic function.”

Recall that Pythagoras theorem Consider

  $h=\sqrt{p^2+b^2}$ ,where h is height p is perpendicular , b is base.

  $8=\sqrt{p^2+4^2}$

  $\begin{array}{l}64=p^2+16\\ p^2=64-16\\ p^2=48\\ p=\sqrt{48}\end{array}$

Perpendicular p=

  $\sqrt{48}$

Recall the trigonometric function

  $\sin \theta =\frac{perpendicular}{height}$ = $\frac{\sqrt{48}}{8}=0.866$

Therefore,

  $\sin \theta =0.866$

  $\cos \theta =\frac{base}{height}$ = $\cos \theta =\frac{4}{8}=0.50$

Therefore,

  $\cos \theta =0.50$

  $\tan \theta =\frac{perpendicular}{base}$

  $=\frac{\sqrt{48}}{4}=1.732$

Therefore,

  $\tan \theta =1.732$

  $\cos ec\theta =\frac{1}{\sin \theta }$

  $=\frac{8}{\sqrt{48}}=1.154$

Therefore, $\cos ec\theta =1.154$

  $\sec \theta =\frac{1}{\cos \theta }$

  $=\frac{8}{4}=2$

Therefore, $\sec \theta =2$

  $\cot \theta =\frac{1}{\tan \theta }$

  $=\frac{4}{\sqrt{48}}=0.577$

So, the third side is $h=\sqrt{48}$ and trigonometric function

  $\begin{array}{l}\sin \theta =0.866\\ \cos \theta =0.50\\ \tan \theta =1.732\\ \cos ec\theta =1.154\\ \sec \theta =2\\ \cot \theta =0.577\end{array}$ where h is height,p is perpendicular, b is base.