Chapter 12.1, Problem 14PPS

To determine

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

Answer to Problem 14PPS

The all the values of trigonometric functions are $\sin \theta =\frac{9}{41},\cos \theta =\frac{40}{41},\tan \theta =\frac{9}{40},\csc \theta =\frac{41}{9},\sec \theta =\frac{41}{40},\cot \theta =\frac{40}{9}$ .

Explanation of Solution

Given:

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

  

Concept Used:

In this, use the concept of pythagoras 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 $p=9,b=40$ .

  $\begin{array}{l}{h}^2=p^2+b^2\\ h^2=9^2+{40}^2\\ h^2=81+1600\\ h^2=1681\\ h=\sqrt{1681}=41\end{array}$

Now, use the trigonometric ratios:

  $\sin \theta =\frac{9}{41},\cos \theta =\frac{40}{41},\tan \theta =\frac{9}{40},\csc \theta =\frac{41}{9},\sec \theta =\frac{41}{40},\cot \theta =\frac{40}{9}$

Conclusion:

The all the values are $\sin \theta =\frac{9}{41},\cos \theta =\frac{40}{41},\tan \theta =\frac{9}{40},\csc \theta =\frac{41}{9},\sec \theta =\frac{41}{40},\cot \theta =\frac{40}{9}$ .