Chapter 6.3, Problem 30E

To determine

To calculate: The value of the trigonometric function $\csc \frac{5\pi }{4}$ .

Answer to Problem 30E

The value of the trigonometric function $\csc \frac{5\pi }{4}$ is $-\sqrt{2}$ .

Explanation of Solution

Given information:

The trigonometric function $\csc \frac{5\pi }{4}$ .

Formula used:

Coordinate plane is divided into four quadrants.

In the first quadrant all trigonometric functions that is $\sin \theta ,cos\theta ,\tan \theta ,\csc \theta ,\sec \theta \text{ and }\cot \theta$ are positive.

In the second quadrant only sine and cosecant trigonometric functions that is $\sin \theta \text{ and }\csc \theta$ are positive and remaining $cos\theta ,\tan \theta ,\sec \theta \text{ and }\cot \theta$ are negative.

In the third quadrant only tangent and cotangent trigonometric functions that is $\tan \theta \text{ and }\cot \theta$ are positive and remaining $cos\theta ,sin\theta ,\sec \theta \text{ and }\csc \theta$ are negative.

In the fourth quadrant only cosine and secant trigonometric functions that is $\cos \theta \text{ and }\sec \theta$ are positive and remaining $\sin \theta ,\tan \theta ,\csc \theta \text{ and }\cot \theta$ are negative.

The reference angle $\overline{\theta }$ is the angle associated with the angle $\theta$ in the standard position. It is the acute angle that is formed by the terminal side of $\theta$ and the x -axis.

Calculation:

Consider the provided trigonometric function $\csc \frac{5\pi }{4}$ .

Denote,

  $\begin{array}{c}\theta =\frac{5\pi }{4}\\ \theta =\frac{5\pi }{4}\cdot \frac{180}{\pi }\\ \theta ={225}^{\circ }\end{array}$ .

First estimate the reference angle $\overline{\theta }$ associated with the angle $\theta$ .

From the figure provided below,

  

The provided angle has terminal side in Quadrant $\text{III}$ .

The reference angle is,

  $\begin{array}{c}\overline{\theta }={225}^{\circ }-{180}^{\circ }\\ ={45}^{\circ }\end{array}$

Recall that coordinate plane is divided into four quadrants.

In the third quadrant only tangent and cotangent trigonometric functions that is $\tan \theta \text{ and }\cot \theta$ are positive and remaining $cos\theta ,sin\theta ,\sec \theta \text{ and }\csc \theta$ are negative.

So, cosecant trigonometric function, $\csc \frac{5\pi }{4}$ is negative.

Therefore,

  $\begin{array}{c}\csc \frac{5\pi }{4}=-\csc \frac{\pi }{4}\\ =-\sqrt{2}\end{array}$

Thus, the value of the trigonometric function $\csc \frac{5\pi }{4}$ is $-\sqrt{2}$ .