Chapter 8, Problem 1RE

To determine

To find: The values of six trigonometric function of a given angle $\theta$ .

Answer to Problem 1RE

The values of six trigonometric functions are $\sin \theta =\frac{4}{5}$ , $\cos \theta =\frac{3}{5}$ , $\tan \theta =\frac{4}{3}$ , $\csc \theta =\frac{5}{4}$ , $\sec \theta =\frac{5}{3}$ , $\cot \theta =\frac{3}{4}$ .

Explanation of Solution

Given information:

The opposite side of the angle $\theta =4$ .

The adjacent side of the angle $\theta =3$ .

Concept used:

The Pythagorus theorem is given by:

  ${\left(\text{Hypotenuse}\right)}^2={\left(\text{Perpendicular}\right)}^2+{\left(\text{Base}\right)}^2$ .

The value of $\text{Hypotenuse}$ is calculated as,

  $\begin{array}{c}\text{Hypotenuse}=\sqrt{4^2+3^2}\\ =\sqrt{25}\\ =5\end{array}$

To calculate the value of six trigonometric functions further simplified as,

  $\begin{array}{c}\sin \theta =\frac{\text{Opposite}}{\text{Hypotenuse}}\\ =\frac{4}{5}\\ \cos \theta =\frac{\text{Adjacent}}{\text{Hypotenuse}}\\ =\frac{3}{5}\end{array}$

Further simplified as:

  $\begin{array}{c}\tan \theta =\frac{\text{Opposite}}{\text{Adjacent}}\\ =\frac{4}{3}\\ \csc \theta =\frac{1}{\sin \theta }\\ =\frac{5}{4}\end{array}$

For further simplified as:

  $\begin{array}{c}\sec \theta =\frac{1}{\cos \theta }\\ =\frac{5}{3}\\ Cot\theta =\frac{1}{\tan \theta }\\ =\frac{3}{4}\end{array}$

Therefore, the value of six trigonometric functions are: $\sin \theta =\frac{4}{5}$ , $\cos \theta =\frac{3}{5}$ , $\tan \theta =\frac{4}{3}$ , $\csc \theta =\frac{5}{4}$ , $\sec \theta =\frac{5}{3}$ , $\cot \theta =\frac{3}{4}$ .