Chapter 1.6, Problem 37E

To determine

To find: The values of six trigonometric functions.

Answer to Problem 37E

The values of six trigonometry are $\sin \theta =\frac{8}{17}$ , $\cos \theta =\frac{15}{17}$ , $\tan \theta =\frac{8}{15}$ , $\cot \theta =\frac{15}{8}$ , $\csc \theta =\frac{17}{8}$ and $\sec \theta =\frac{17}{15}$ .

Explanation of Solution

Given information: The given value is $\theta ={\sin }^{-1}\left(\frac{8}{17}\right)$ .

Formula used: The six trigonometric functions with an angle $\theta$ can be calculated as:

  $\sin \theta =\frac{y}{r}$ , $\cos \theta =\frac{x}{r}$ , $\tan \theta =\frac{y}{x}$ , $\cot \theta =\frac{x}{y}$ , $\csc \theta =\frac{r}{y}$ , $\sec \theta =\frac{r}{x}$

Calculation:

Consider the given angle and simplify.

  $\begin{array}{c}\theta ={\sin }^{-1}\left(\frac{8}{17}\right)\\ \sin \theta =\sin \left({\sin }^{-1}\left(\frac{8}{17}\right)\right)\\ \sin \theta =\frac{8}{17}\end{array}$

Compare the value of $\sin \theta$ with $\sin \theta =\frac{y}{r}$ to get,

  $\begin{array}{c}y=8\\ r=17\end{array}$

Now, by Pythagorean theorem

  $\begin{array}{c}x=\sqrt{{17}^2-8^2}\\ =\sqrt{225}\\ =15\end{array}$

The remaining trigonometric ratios are:

  $\begin{array}{c}\cos \theta =\frac{15}{17}\\ \tan \theta =\frac{8}{15}\\ \cot \theta =\frac{15}{8}\\ \sec \theta =\frac{17}{15}\\ \csc \theta =\frac{17}{8}\end{array}$

Therefore, the values of six trigonometry are $\sin \theta =\frac{8}{17}$ , $\cos \theta =\frac{15}{17}$ , $\tan \theta =\frac{8}{15}$ , $\cot \theta =\frac{15}{8}$ , $\csc \theta =\frac{17}{8}$ and $\sec \theta =\frac{17}{15}$ .