Question

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

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

Short 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-by-step solution

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 θ.

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

These are the exact values of the remaining trigonometric functions.