Chapter 12.1, Problem 13PPS

To determine

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

Answer to Problem 13PPS

The all the values of trigonometric functions are $\sin \theta =\frac{12}{13},\cos \theta =\frac{5}{13},\tan \theta =\frac{12}{5},\csc \theta =\frac{13}{12},\sec \theta =\frac{13}{5},\cot \theta =\frac{5}{12}$ .

Explanation of Solution

Given:

It is given in the question that the figure is given below in which p and h 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 $p=12,h=13$ .

  $\begin{array}{l}{h}^2=p^2+b^2\\ {13}^2={12}^2+b^2\\ b^2=169-144\\ b^2=25\\ b=\sqrt{25}=5\end{array}$

Now, use the trigonometric ratios:

  $\sin \theta =\frac{12}{13},\cos \theta =\frac{5}{13},\tan \theta =\frac{12}{5},\csc \theta =\frac{13}{12},\sec \theta =\frac{13}{5},\cot \theta =\frac{5}{12}$

Conclusion:

The all the values are $\sin \theta =\frac{12}{13},\cos \theta =\frac{5}{13},\tan \theta =\frac{12}{5},\csc \theta =\frac{13}{12},\sec \theta =\frac{13}{5},\cot \theta =\frac{5}{12}$ .