Chapter 0.6, Problem 38E

To determine

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

Answer to Problem 38E

The six trigonometric functions are $\sin \theta =-\frac{5}{13}$ , $\cos \theta =\frac{12}{13}$ , $\tan \theta =-\frac{5}{12}$ , $\csc \theta =-\frac{13}{5}$ , $\sec \theta =\frac{13}{12}$ and $\cot \theta =-\frac{12}{5}$ .

Explanation of Solution

Given information: The given angle is $\theta ={\tan }^{-1}\left(-\frac{5}{12}\right)$ .

Calculation:

Simplify the given angle.

  $\begin{array}{c}\theta ={\tan }^{-1}\left(-\frac{5}{12}\right)\\ \tan \theta =-\frac{5}{12}\end{array}$

Compare the above angle with $\tan \theta =\frac{y}{x}$ to get $x=12$ and $y=-5$ .

The angle $\theta$ lies in the fourth quadrant.

  

Figure (1)

Calculate the value of $x$ .

  $\begin{array}{c}{r}^2={\left(12\right)}^2+{\left(-5\right)}^2\\ r^2=144+25\\ r^2=169\\ r=13\end{array}$

Calculate the trigonometric function sine.

  $\begin{array}{c}\sin \theta =\frac{y}{r}\\ =-\frac{5}{13}\end{array}$

Calculate the trigonometric function cosine.

  $\begin{array}{c}\cos \theta =\frac{x}{r}\\ =\frac{12}{13}\end{array}$

Calculate the trigonometric function cosecant.

  $\begin{array}{c}\csc \theta =\frac{r}{y}\\ =-\frac{13}{5}\end{array}$

Calculate the trigonometric function secant.

  $\begin{array}{c}\sec \theta =\frac{r}{x}\\ =\frac{13}{12}\end{array}$

Calculate the trigonometric function cotangent.

  $\begin{array}{c}\cot \theta =\frac{x}{y}\\ =-\frac{12}{5}\end{array}$

Thus, the six trigonometric functions are $\sin \theta =-\frac{5}{13}$ , $\cos \theta =\frac{12}{13}$ , $\tan \theta =-\frac{5}{12}$ , $\csc \theta =-\frac{13}{5}$ , $\sec \theta =\frac{13}{12}$ and $\cot \theta =-\frac{12}{5}$ .