Chapter 7.4, Problem 3CFU

a.

To determine

To name: the quadrant where $2x$ lies.

Answer to Problem 3CFU

Either third or fourth quadrant

Explanation of Solution

Given information:

Second quadrant angle is $x$ .

Calculation:

It is given that $x$ lies in second quadrant i.e. from $\left(\frac{\pi }{2}+2\pi k\right)$ to $\left(\pi +2\pi k\right)$ .

  $\begin{array}{l}\frac{\pi }{2}+2\pi k\le x\le 2\pi +2\pi k\\ \pi +4\pi k\le 2x\le 4\pi +4\pi k\left[\text{Multiply by 2}\right]\end{array}$

Thus $2x$ lies in the third or fourth quadrant.

b.

To determine

To name: the quadrant where $x$ lies.

Answer to Problem 3CFU

Either second or third quadrant

Explanation of Solution

Given information:

First quadrant angle is $\frac{x}{2}$ .

Calculation:

It is given that $\frac{x}{2}$ lies in first quadrant i.e. from $2\pi k+0$ to $\frac{\pi }{2}+2\pi k$ .

  $\begin{array}{l}2\pi k+0\le \frac{x}{2}\le \frac{\pi }{2}+2\pi k\\ 4\pi k+0\le x\le \pi +4\pi k\left[\text{Multiply by 2}\right]\end{array}$

Thus $x$ lies in the second or third quadrant.

c.

To determine

To name: the quadrant where $\frac{x}{2}$ lies.

Answer to Problem 3CFU

First quadrant

Explanation of Solution

Given information:

Second quadrant angle is $2x$ .

Calculation:

It is given that $2x$ lies in second quadrant i.e. between $\frac{\pi }{2}$ and $\pi$ .

  $\begin{array}{l}\frac{\pi }{2}<2x<\pi \\ \frac{\pi }{8}<\frac{x}{2}<\frac{\pi }{4}\left[\text{Divide each term by 4}\right]\end{array}$

Thus $\frac{x}{2}$ lies in the first quadrant.