Q. 19 - Precalculus Enhanced with Graphing Utilities

Question

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

$\sin \theta =\frac{12}{13},\theta$lies in the quadrant ll.

Step-by-Step Solution

Answer

The exact values of the remaining trigonometric functions of $\sin \theta =\frac{12}{13}$ are,

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

Step 1 Given expression is,

$\sin \theta =\frac{12}{13}$ and $\theta$ lies in the second quadrant.

So,$\sin \theta , csc \theta$ are positive in the second quadrant and the remaining trigonometric functions are negative.

Let $\alpha$ be the reference angle for $\theta$.

$\sin \theta =\sin \alpha$.

So,$\sin \theta =\frac{12}{13}=\frac{opposite}{hypotenuse}$

Use $b=12,r=13$ to construct a triangle.

From the figure, we obtain $a=5$.

Step 2 Now calculate the remaining trigonometric functions.

The exact values of the remaining trigonometric functions of the reference angle $\alpha$ are,

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

Step 3 The exact values of the remaining trigonometric functions of the reference angle θ are given as,

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

These are the remaining trigonometric functions.