Chapter 0.6, Problem 37E

To determine

To calculate: The six trigonometric functions at angle $\theta$ .

Answer to Problem 37E

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

Explanation of Solution

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

Calculation:

Simplify the given angle.

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

Compare the above angle with $\sin \theta =\frac{y}{r}$ to get $y=8$ and $r=17$ .

The angle $\theta$ lies in the first quadrant, because sine is positive.

  

Figure (1)

Calculate the value of $x$ .

  $\begin{array}{c}{\left(17\right)}^2=x^2+{\left(8\right)}^2\\ 289=x^2+64\\ 225=x^2\\ x=15\end{array}$

Calculate the trigonometric function cosine.

  $\begin{array}{c}\cos \theta =\frac{x}{r}\\ =\frac{15}{17}\end{array}$

Calculate the trigonometric function tangent.

  $\begin{array}{c}\tan \theta =\frac{y}{x}\\ =\frac{8}{15}\end{array}$

Calculate the trigonometric function cosecant.

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

Calculate the trigonometric function secant.

  $\begin{array}{c}\sec \theta =\frac{r}{x}\\ =\frac{17}{15}\end{array}$

Calculate the trigonometric function cotangent.

  $\begin{array}{c}\cot \theta =\frac{x}{y}\\ =\frac{15}{8}\end{array}$

Thus, the six trigonometric functions are $\sin \theta =\frac{8}{17}$ , $\cos \theta =\frac{15}{17}$ , $\tan \theta =\frac{8}{17}$ , $\csc \theta =\frac{17}{8}$ , $\sec \theta =\frac{17}{15}$ and $\cot \theta =\frac{15}{8}$ .