Chapter 4.4, Problem 65E

To determine

The sine, cosine, and tangent of the angle $\frac{11\pi }{4}$ without using a calculator.

Answer to Problem 65E

The sine, cosine, and tangent of the angle $\frac{11\pi }{4}$ without using a calculator is

  $\begin{array}{l}\sin \frac{11\pi }{4}=\frac{1}{\sqrt{2}}\\ \cos \frac{11\pi }{4}=-\frac{1}{\sqrt{2}}\\ \tan \frac{11\pi }{4}=-1\end{array}$

Explanation of Solution

Given info:

  $\theta =\frac{11\pi }{4}$

Formula used:

Let be $\theta$ an angle in standard position. Its reference angle $\theta \text{'}$ is the acute angleformed by the terminal side of $\theta$ and the horizontal axis.

In quadrant I all trigonometric functions are positive.

In quadrant II sine functions are positive.

In quadrant III tangent functions are positive.

In quadrant IV cosine functions are positive.

Calculation:

We have

  $\begin{array}{l}\theta =\frac{11\pi }{4}=2\pi +\frac{3\pi }{4}\\ \theta =\frac{3\pi }{4}\\ \theta \text{'}=\pi -\frac{3\pi }{4}=\frac{\pi }{4}\end{array}$

Here $\theta =\frac{3\pi }{4}$ lies in second quadrant,

In quadrant II sine functions are positive.

Hence

  $\begin{array}{l}\sin \frac{11\pi }{4}=\sin \frac{\pi }{4}=\frac{1}{\sqrt{2}}\\ \cos \frac{11\pi }{4}=-\cos \frac{\pi }{4}=-\frac{1}{\sqrt{2}}\\ \tan \frac{11\pi }{4}=-\tan \frac{\pi }{4}=-1\end{array}$

Conclusion:

Thus,sine, cosine, and tangent of the angle $\frac{11\pi }{4}$ without using a calculator is

  $\begin{array}{l}\sin \frac{11\pi }{4}=\frac{1}{\sqrt{2}}\\ \cos \frac{11\pi }{4}=-\frac{1}{\sqrt{2}}\\ \tan \frac{11\pi }{4}=-1\end{array}$