Chapter 4.4, Problem 11E

(a)

To determine

To find: The exact value of six trigonometric function of each angle $\theta$ .

Answer to Problem 11E

The exact value of six trigonometric function of each angle $\theta$ are $\cos \theta =\frac{4}{5},\sin \theta =\frac{3}{5},\cot \theta =\frac{4}{3},\tan \theta =\frac{3}{4},$$\sec \theta =\frac{5}{4},\csc \theta =\frac{5}{3}$ .

Explanation of Solution

Given information:

The given coordinates are shown in figure (1).

  

Figure (1)

Calculation:

Use $\theta$ be an angle in standard position with $\left(x,y\right)$ from figure (1).

  $\begin{array}{c}r=\sqrt{x^2+y^2}\\ \ne 0\end{array}$

The trigonometric functions of any angle.

  $\begin{array}{l}\cos \theta =\frac{x}{r}\\ \sin \theta =\frac{y}{r}\\ \cot \theta =\frac{x}{y}\\ \tan \theta =\frac{y}{x}\end{array}$

  $\begin{array}{l}\sec \theta =\frac{r}{x}\\ \csc \theta =\frac{r}{y}\end{array}$

Calculate the value of $r$ and then calculate the value of the trigonometric functions.

  $\begin{array}{c}r=\sqrt{x^2+y^2}\\ =\sqrt{4^2+3^2}\\ =5\end{array}$

Substitute $4$ for $x$ , $3$ for $y$ , and $5$ for $r$ in trigonometric functions.

  $\begin{array}{l}\cos \theta =\frac{4}{5}\\ \sin \theta =\frac{3}{5}\\ \cot \theta =\frac{4}{3}\\ \tan \theta =\frac{3}{4}\end{array}$

  $\begin{array}{l}\sec \theta =\frac{5}{4}\\ \csc \theta =\frac{5}{3}\end{array}$

Therefore, the exact value of six trigonometric function of each angle $\theta$ are $\cos \theta =\frac{4}{5},\sin \theta =\frac{3}{5},\cot \theta =\frac{4}{3},\tan \theta =\frac{3}{4},$$\sec \theta =\frac{5}{4},\csc \theta =\frac{5}{3}$ .

(b)

To determine

To find: The exact value of six trigonometric function of each angle $\theta$ .

Answer to Problem 11E

The exact value of six trigonometric function of each angle $\theta$ are $\cos \theta =-\frac{8}{17},\sin \theta =\frac{15}{17},\cot \theta =-\frac{8}{15},\tan \theta =-\frac{15}{8},$$\sec \theta =-\frac{17}{8},\csc \theta =\frac{17}{15}$ .

Explanation of Solution

Given information:

The given coordinates are shown in figure (2).

  

Figure (2)

Calculation:

Use $\theta$ be an angle in standard position with $\left(x,y\right)$ from figure (2).

  $\begin{array}{c}r=\sqrt{x^2+y^2}\\ \ne 0\end{array}$

The trigonometric functions of any angle.

  $\begin{array}{l}\cos \theta =\frac{x}{r}\\ \sin \theta =\frac{y}{r}\\ \cot \theta =\frac{x}{y}\\ \tan \theta =\frac{y}{x}\end{array}$

  $\begin{array}{l}\sec \theta =\frac{r}{x}\\ \csc \theta =\frac{r}{y}\end{array}$

Calculate the value of $r$ and then calculate the value of the trigonometric functions.

  $\begin{array}{c}r=\sqrt{x^2+y^2}\\ =\sqrt{{\left(-8\right)}^2+{15}^2}\\ =17\end{array}$

Substitute $-8$ for $x$ , $15$ for $y$ , and $17$ for $r$ in trigonometric functions.

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

  $\begin{array}{l}\sec \theta =-\frac{17}{8}\\ \csc \theta =\frac{17}{15}\end{array}$

Therefore, the exact value of six trigonometric function of each angle $\theta$ are $\cos \theta =-\frac{8}{17},\sin \theta =\frac{15}{17},\cot \theta =-\frac{8}{15},\tan \theta =-\frac{15}{8},$$\sec \theta =-\frac{17}{8},\csc \theta =\frac{17}{15}$ .