Chapter 13.2, Problem 58STP

To determine

Calculate the values of six trigonometric functions for angle $\theta$ .

Answer to Problem 58STP

The all the values of trigonometric functions are $\sin \theta =\frac{3}{\sqrt{205}},\cos \theta =\frac{14}{\sqrt{205}},\tan \theta =\frac{3}{14},\csc \theta =\frac{\sqrt{205}}{3},\sec \theta =\frac{\sqrt{205}}{14},\cot \theta =\frac{14}{3}$ .

Explanation of Solution

Given:

It is given in the question that the figure is given below in which b and p are given:

  

Concept Used:

In this, use the concept of pythgoras theorem and trigonometric ratios and the formula will be $h^2=p^2+b^2$ , $\sin \theta =\frac{p}{h},\cos \theta =\frac{b}{h},\tan \theta =\frac{p}{b},\csc \theta =\frac{h}{p},\sec \theta =\frac{h}{b},\cot \theta =\frac{b}{p}$ .

Calculation: Here, use Pythagoras theorem where $b=14,p=3$ .

  $\begin{array}{l}{h}^2=p^2+b^2\\ h^2=3^2+{14}^2\\ h^2=9+196\\ h^2=205\\ h=\sqrt{205}\end{array}$

Now, use the trigonometric ratios:

  $\sin \theta =\frac{3}{\sqrt{205}},\cos \theta =\frac{14}{\sqrt{205}},\tan \theta =\frac{3}{14},\csc \theta =\frac{\sqrt{205}}{3},\sec \theta =\frac{\sqrt{205}}{14},\cot \theta =\frac{14}{3}$

Conclusion:

The all the values are $\sin \theta =\frac{3}{\sqrt{205}},\cos \theta =\frac{14}{\sqrt{205}},\tan \theta =\frac{3}{14},\csc \theta =\frac{\sqrt{205}}{3},\sec \theta =\frac{\sqrt{205}}{14},\cot \theta =\frac{14}{3}$ .