Chapter 4.4, Problem 29E

To determine

To find: The values of the six trigonometric functions of $\theta$ if $\tan \theta =\frac{-15}{8}$ and $\sin \theta <0$ .

Answer to Problem 29E

The values of the six trigonometric functions of $\theta$ are as follows:

  $\begin{array}{l}\sin \theta =\frac{-15}{17},\cos \theta =\frac{8}{17},\tan \theta =\frac{-15}{8}\\ \cos ec\theta =\frac{-17}{15},\sec \theta =\frac{17}{8},\cot \theta =\frac{-8}{15}\end{array}$

Explanation of Solution

Given information:

  $\tan \theta =\frac{-15}{8}$ and $\sin \theta <0$ .

Calculation:

Since, $\sin \theta <0$ in the third and the fourth quadrant. It is given that $\tan \theta <0$ . $\tan \theta <0$ inthe second and the fourth quadrant. Hence, $\theta$ lies in the fourth quadrant.

In the fourth quadrant, $\cos \theta >0$ and $\sec \theta >0$ only whereas $\tan \theta <0,\cot \theta <0,\sin \theta <0$ and $co\sec \theta <0.$

Since, $\tan \theta =\frac{-15}{8}$ lies in the fourth quadrant. Let us suppose point A $\left(8,-15\right)$ is drawn on the line with angle $\theta$ .

  

From the co-ordinate plane, the base is $8$ and perpendicular is $-15$ and the hypotenuse of the triangle is $r$ .

Apply Pythagoras Theorem to calculate the hypotenuse $r$ .

  $\begin{array}{c}r=\sqrt{{\left(x\right)}^2+{\left(y\right)}^2}\\ r=\sqrt{{\left(8\right)}^2+{\left(-15\right)}^2}\\ r=\sqrt{64+225}\\ =\sqrt{289}\\ =17\end{array}$

Calculate the trigonometric functions.

  $\begin{array}{c}\sin \theta =\frac{y}{r}\\ =\frac{-15}{17}\end{array}$

and

  $\begin{array}{c}\cos \theta =\frac{x}{r}\\ =\frac{8}{17}\end{array}$

and

  $\begin{array}{c}\tan \theta =\frac{y}{x}\\ =\frac{-15}{8}\end{array}$

and

  $\begin{array}{c}\text{cosec}\theta =\frac{1}{\sin \theta }\\ =\frac{17}{-15}\end{array}$

and

  $\begin{array}{c}\sec \theta =\frac{1}{\cos \theta }\\ =\frac{17}{8}\end{array}$

and

  $\begin{array}{c}\text{cot}\theta =\frac{1}{\tan \theta }\\ =\frac{-8}{15}\end{array}$

Therefore, the exact values of six trigonometric functions are as follows.

  $\begin{array}{l}\sin \theta =\frac{-15}{17},\cos \theta =\frac{8}{17},\tan \theta =\frac{-15}{8}\\ \cos ec\theta =\frac{-17}{15},\sec \theta =\frac{17}{8},\cot \theta =\frac{-8}{15}\end{array}$