Chapter 1, Problem 57RE

To determine

To find: all six trigonometric values of $\theta$ with the condition $\theta ={\cos }^{-1}\left(3/7\right)$ .

Answer to Problem 57RE

All the trigonometric functions are $\sin \theta =\frac{2\sqrt{10}}{7}$ , $\cos \theta =\frac{3}{7}$ , $\tan \theta =\frac{2\sqrt{10}}{3}$ , $\cot \theta =\frac{3}{2\sqrt{10}}$ , $\sec \theta =\frac{7}{3}$ and $\text{cosec}\theta =\frac{7}{2\sqrt{10}}$ .

Explanation of Solution

Given information:

The given condition is $\theta ={\cos }^{-1}\left(3/7\right)$ .

Calculation:

Solve the given condition.

  $\begin{array}{c}\theta ={\cos }^{-1}\left(3/7\right)\\ \cos \theta =\frac{3}{7}\end{array}$

The value of sine function can be calculated as:

  $\begin{array}{c}\sin \theta =\sqrt{1-{\cos }^2\theta }\\ =\sqrt{1-{\left(\frac{3}{7}\right)}^2}\\ =\sqrt{1-\frac{9}{49}}\\ =\frac{2\sqrt{10}}{7}\end{array}$

The value of tangent function can be calculated as:

  $\begin{array}{c}\tan \theta =\frac{\sin \theta }{\cos \theta }\\ =\frac{2\sqrt{10}/7}{3/7}\\ =\frac{2\sqrt{10}}{3}\end{array}$

The value of cotangent function can be calculated as:

  $\begin{array}{c}\cot \theta =\frac{1}{\tan \theta }\\ =\frac{1}{2\sqrt{10}/3}\\ =\frac{3}{2\sqrt{10}}\end{array}$

The value of secant function can be calculated as:

  $\begin{array}{c}\sec \theta =\frac{1}{\cos \theta }\\ =\frac{1}{3/7}\\ =\frac{7}{3}\end{array}$

The value of cosec function can be calculated as:

  $\begin{array}{c}\text{cosec}\theta =\frac{1}{\sin \theta }\\ =\frac{1}{2\sqrt{10}/7}\\ =\frac{7}{2\sqrt{10}}\end{array}$

Therefore, all the trigonometric functions are $\sin \theta =\frac{2\sqrt{10}}{7}$ , $\cos \theta =\frac{3}{7}$ , $\tan \theta =\frac{2\sqrt{10}}{3}$ , $\cot \theta =\frac{3}{2\sqrt{10}}$ , $\sec \theta =\frac{7}{3}$ and $\text{cosec}\theta =\frac{7}{2\sqrt{10}}$ .