Q. 18 - Precalculus Enhanced with Graphing Utilities

Question

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

$sec \theta =-\frac{5}{4},\tan \theta <0$

Step-by-Step Solution

Answer

The exact value of the remaining trigonometric functions of $sec \theta =-\frac{5}{4}$ are,

$$\begin{gathered} \sin \theta =-\frac{3}{5} \\ \cos \theta =\frac{4}{5} \\ \tan \theta =-\frac{3}{4} \\ csc \theta =-\frac{5}{3} \\ cot \theta =-\frac{4}{3} \end{gathered}$$

Step 1 Given expression is,

$sec \theta =-\frac{5}{4}$,$\tan \theta <0$.

Since $\theta$ lies in the second quadrant, $tan \theta , sec \theta$ are negative.

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

So, $sec \alpha =\frac{5}{4}=\frac{Hypotenuse}{Adjacent }$

Step 2 use the values r = 5 , a = 4 to construct a triangle.

From the figure, we obtain $b=3$.

Step 3 Now calculate the remaining trigonometric functions.

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

$$\begin{gathered} \sin \alpha =\frac{b}{r} \\ =\frac{3}{5} \\ \cos \alpha =\frac{a}{r} \\ =\frac{4}{5} \\ \tan \alpha =\frac{b}{a} \\ =\frac{4}{3} \\ csc \alpha =\frac{r}{b} \\ =\frac{5}{3} \\ cot \alpha =\frac{a}{b} \\ =\frac{4}{3} \end{gathered}$$

Step 4 Now calculate the remaining trigonometric functions of the angle &#952; .

$$\begin{gathered} \sin \theta =-\frac{3}{5} \\ \cos \theta =\frac{4}{5} \\ \tan \theta =-\frac{3}{4} \\ csc \theta =-\frac{5}{3} \\ cot \theta =-\frac{4}{3} \end{gathered}$$

These are the exact values of the remaining trigonometric functions.