Chapter 1, Problem 1RE

Using Radian or Degree Measure In Exercises 1-4, (a) sketch the angle in standard position, (b) determine the quadrant in which the angle lies, and (c) determine two coterminal angles (one positive and one negative).

$\frac{15\pi }{4}$

To determine

To graph: The angle, $\frac{15\pi }{4}$, in standard position.

Explanation of Solution

Graph:

Consider, the provided angle, $\frac{15\pi }{4}$,

Now, in order to sketch the angle, $\frac{15\pi }{4}$, first find the quadrant in which, the angle, $\frac{15\pi }{4}$, lies.

Since, the angle, $\frac{15\pi }{4}$, is between the interval, $\frac{3\pi }{2}<\frac{15\pi }{4}<2\pi$, which, lies in the $\text{quadrant ΙV}$.

Therefore, $\frac{15\pi }{4}$ lies in the fourth quadrant and has a positive sign as, it rotates in a counterclockwise direction, as shown in the figure below.

To determine

The quadrant in which, the angle, $\frac{15\pi }{4}$, lies.

Answer to Problem 1RE

The quadrant in which, the angle, $\frac{15\pi }{4}$, lies is $\text{Quadrant ΙV}$.

Explanation of Solution

Consider, the provided angle,

$\frac{15\pi }{4}$

The co-ordinate system consists of four quadrants numbered as $\text{Ι, ΙΙ, ΙΙΙ and ΙV}$. The angles between $0$ and $2\pi$ lies in each of the four quadrants.

Quadrant $\text{Ι}$ : The measure of angles lies between the interval, $0<\theta <\frac{\pi }{2}$.

Quadrant $\text{ΙΙ}$ : The measure of angles lies between the interval, $\frac{\pi }{2}<\theta <\pi$.

Quadrant $\text{ΙΙΙ}$ : The measure of angles lies between the interval, $\pi <\theta <\frac{3\pi }{2}$.

Quadrant $\text{ΙV}$ : The measure of angles lies between the interval, $\frac{3\pi }{2}<\theta <2\pi$.

Since, the angle $\frac{15\pi }{4}$ is between the interval, $\frac{3\pi }{2}<\frac{15\pi }{4}<2\pi$.

Therefore, the angle, $\frac{15\pi }{4}$, lies in the fourth quadrant.

Hence, the quadrant in which, the angle, $\frac{15\pi }{4}$, lies is $\text{Quadrant ΙV}$.

To determine

The (one positive and one negative) co-terminal angle, for the angle, $\frac{15\pi }{4}$.

Answer to Problem 1RE

The (one positive and one negative) co-terminal angle, for the angle, $\frac{15\pi }{4}$ are $\frac{7\pi }{4}\text{ and }-\frac{\pi }{4}$.

Explanation of Solution

Co-terminal angle:

If two angles are made by the same initial and terminal side, then, such angles are called co-terminal angles. The positive and negative co-terminal angle of an angle can be obtained by subtracting or adding $2\pi$ in the angle.

Consider, the provided angle,

$\frac{15\pi }{4}$

Now, in order to get the positive and negative co-terminal angles, add and subtract $2\pi$ from the $\frac{15\pi }{4}$ that is,

For the positive angle, $\frac{15\pi }{4}$, subtract $2\pi$ to obtain a coterminal angle,

$\begin{array}{c}\theta =\frac{15\pi }{4}-2\pi \\ =\frac{15\pi -8\pi }{4}\\ =\frac{7\pi }{4}\end{array}$

And,

For the negative angle, $\frac{15\pi }{4}$, subtract $2\pi$ to obtain a coterminal angle,

$\begin{array}{c}\theta =\frac{7\pi }{4}-2\pi \\ =\frac{7\pi -8\pi }{4}\\ =-\frac{\pi }{4}\end{array}$

Hence, the (one positive and one negative) co-terminal angle, for the angle, $\frac{15\pi }{4}$, are $\frac{7\pi }{4}\text{ and }-\frac{\pi }{4}$.