Chapter C, Problem 12E

To determine

To calculate:

The value of the exact trigonometric ratio of the given radian angle:

  $\frac{4\pi }{3}$

Answer to Problem 12E

The exact trigonometric ratio of the $\frac{4\pi }{3}$

  $\begin{array}{l}\sin \frac{4\pi }{3}=\frac{-\sqrt{3}}{2}\\ \cos ec\frac{4\pi }{3}=\frac{2}{-\sqrt{3}}\\ \cos \frac{4\pi }{3}=\frac{-1}{2}\\ \sec \frac{4\pi }{3}=-2\\ \tan \frac{4\pi }{3}=\cot \frac{4\pi }{3}=\sqrt{3}\end{array}$

Explanation of Solution

Given information:

  $\frac{4\pi }{3}$

Formula Used:

Trigonometric ratios are given by the following formula:

  $\begin{array}{l}\sin \theta =\frac{y}{r}\cos ec\theta =\frac{r}{y}\\ \cos \theta =\frac{x}{r}\sec \theta =\frac{r}{x}\\ \tan \theta =\frac{y}{x}\cot \theta =\frac{x}{y}\end{array}$

Calculation:

The given value of radian is $\frac{4\pi }{3}$ . It is positive angle value that means the rotation is measured from the $x-$ axis as counter-clockwise direction.

Know that:

In one full rotation the angle is $2\pi$ . And the relation between the degree and radian is shown by below expression:

  $\begin{array}{l}\pi rad=180°\\ 1rad=\frac{180°}{\pi }\end{array}$

Now, rewrite the given angle in the standard position:

  $\begin{array}{l}\frac{4\pi }{3}=\frac{3\pi }{3}+\frac{\pi }{3}\\ \frac{4\pi }{3}=\pi +\frac{\pi }{3}\end{array}$

Thus, the given angle $\frac{4\pi }{3}$ measures angle half revolution plus $\frac{\pi }{3}$ from the positive $x-$ axis.

This angle is drawn as below:

  

Here, the triangle is fixed at terminal $\left(x,y\right)=\left(-1,-\sqrt{3}\right)$ and $r=\sqrt{{\left(-1\right)}^2+{\left(-\sqrt{3}\right)}^2}=\sqrt{4}=2$

Therefore, trigonometric ratios are given as:

  $\begin{array}{l}\sin \frac{4\pi }{3}=\frac{y}{r}=\frac{-\sqrt{3}}{2}\\ \cos ec\frac{4\pi }{3}=\frac{r}{y}=\frac{2}{-\sqrt{3}}\\ \cos \frac{4\pi }{3}=\frac{x}{r}=\frac{-1}{2}\\ \sec \frac{4\pi }{3}=\frac{r}{x}=-\frac{2}{1}=-2\\ \tan \frac{4\pi }{3}=\frac{y}{x}=\frac{-\sqrt{3}}{-1}=\sqrt{3}\\ \cot \frac{4\pi }{3}=\frac{x}{y}=\frac{-1}{-\sqrt{3}}=\sqrt{3}\end{array}$