Chapter C, Problem 11E

To determine

To calculate:

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

  $\frac{3\pi }{4}$

Answer to Problem 11E

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

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

Explanation of Solution

Given information:

  $\frac{3\pi }{4}$

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 $-\frac{3\pi }{4}rad$ . It is negative angle value that means the rotation is measured from the $x-$ axis as 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:

  $\begin{array}{l}-\frac{3\pi }{4}rad=-\frac{3\pi }{4}\times \frac{180°}{\pi }\\ -\frac{3\pi }{4}rad=-3\times 45°\\ -\frac{3\pi }{4}rad=-135°\end{array}$

Thus, the given angle $-\frac{3\pi }{4}rad$ measures angle $-135°$ or it is $\frac{3}{4}$ of the half revolution $\left(180°\right)$ from the positive $x-$ axis.

This angle is drawn as below:

  

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

Therefore, trigonometric ratios are given as:

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