Chapter 4.1, Problem 17E

a.

To determine

To find:The complement and supplement of angle $\frac{\pi }{12}$ .

Answer to Problem 17E

The complement and supplement of angle $\frac{\pi }{12}$ are $\frac{5\pi }{12}$ and $\frac{11\pi }{12}$ , respectively.

Explanation of Solution

Given information:

The give angle is $\frac{\pi }{12}$ .

Calculation:

Two angles are complements of each other when their sum is $\frac{\pi }{2}$ . To determine the complement angle, subtract the given angle from $\frac{\pi }{2}$ .

Two angles are supplements of each other when their sum is $\pi$ . To determine the supplement angle, subtract the given angle from $\pi$ .

Calculate the complement angle.

  $\begin{array}{c}\frac{\pi }{2}-\frac{\pi }{12}=\frac{6\pi -\pi }{12}\\ =\frac{5\pi }{12}\end{array}$

Calculate the supplement angle.

  $\begin{array}{c}\pi -\frac{\pi }{12}=\frac{12\pi -\pi }{12}\\ =\frac{11\pi }{12}\end{array}$

Therefore, the complement and supplement of angle $\frac{\pi }{12}$ are $\frac{5\pi }{12}$ and $\frac{11\pi }{12}$ , respectively.

b.

To determine

To find:The complement and supplement of angle $\frac{11\pi }{12}$ .

Answer to Problem 17E

The complement is not possible while the supplement angle is $\frac{\pi }{12}$ .

Explanation of Solution

Given information:

The give angle is $\frac{11\pi }{12}$ .

Calculation:

Two angles are complements of each other when their sum is $\frac{\pi }{2}$ . To determine the complement angle, subtract the given angle from $\frac{\pi }{2}$ .

Two angles are supplements of each other when their sum is $\pi$ . To determine the supplement angle, subtract the given angle from $\pi$ .

Since, $\frac{11\pi }{12}$ is greater than $\frac{\pi }{2}$ and it gives negative value. Complements are positive angles. So, its complement is not possible.

Calculate the supplement angle.

  $\begin{array}{c}\pi -\frac{11\pi }{12}=\frac{12\pi -11\pi }{12}\\ =\frac{\pi }{12}\end{array}$

Therefore, the complement is not possible while the supplement angle is $\frac{\pi }{12}$ .