Question

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

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

Short 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-by-step solution

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,role="math" localid="1646404662147" $\tan \theta =\frac{b}{a}=\tan \alpha$.

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 θ.

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

These are the exact values of the remaining trigonometric functions.