Q. 17 - Precalculus Enhanced with Graphing Utilities

Question

In Problems 16–23, find the exact value of each of the remaining trigonometric functions.

$\tan \theta =\frac{12}{5},\sin \theta <0$.

Step-by-Step Solution

Answer

The exact value of the remaining trigonometric functions for the given expression $\tan \theta =\frac{12}{5}$ are,

$$\begin{gathered} \sin \theta =-\frac{12}{13} \\ \cos \theta =-\frac{5}{13} \\ csc \theta =-\frac{13}{12} \\ sec \theta =-\frac{13}{5} \\ cot \theta =\frac{5}{12} \end{gathered}$$

Step 1 Given expression is,

$\tan \theta =\frac{12}{5},\sin \theta <0$

$\theta$ lies in the third quadrant where $\sin \theta$ is negative and $\tan \theta$ is positive.

Let $\alpha$ be the reference angle for $\theta$ then, $\tan \theta =\frac{b}{a}=\tan \alpha$.

So,$\tan \alpha =\frac{12}{5}=\frac{opp.side}{adj.side}$

Step 2 use the values b = 12 , a = 5 to construct a triangle.

From the figure, we obtain $r=13$.

Step 3 Now write the remaining trigonometric values.

The remaining five trigonometric functions of the reference angle $\alpha$ are calculated as follows,

$$\begin{gathered} \sin \alpha =\frac{b}{r} \\ =\frac{12}{13} \\ \cos \alpha =\frac{a}{r} \\ =\frac{5}{13} \\ csc \alpha =\frac{r}{b} \\ =\frac{13}{12} \\ sec \alpha =\frac{r}{a} \\ =\frac{13}{5} \\ cot \alpha =\frac{a}{b} \\ =\frac{5}{12} \end{gathered}$$

Step 4 Find the value of the remaining trigonometric functions of the angle &#952; .

$$\begin{gathered} \sin \theta =-\frac{b}{r} \\ =-\frac{12}{13} \\ \cos \theta =-\frac{a}{r} \\ =-\frac{5}{13} \\ csc \theta =-\frac{r}{b} \\ =-\frac{13}{12} \\ sec \theta =-\frac{r}{a} \\ =-\frac{13}{5} \\ cot \theta =\frac{a}{b} \\ =\frac{5}{12} \end{gathered}$$

These are the exact values of the remaining trigonometric functions.