Chapter 4.1, Problem 18E

a.

To determine

To find:The complement and supplement of angle $\frac{\pi }{3}$ .

Answer to Problem 18E

The complement and supplement of angle $\frac{\pi }{3}$ are $\frac{\pi }{6}$ and $\frac{2\pi }{3}$ , respectively.

Explanation of Solution

Given information:

The give angle is $\frac{\pi }{3}$ .

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.

  $\begin{array}{c}\frac{\pi }{2}-\frac{\pi }{3}=\frac{3\pi -2\pi }{6}\\ =\frac{\pi }{6}\end{array}$

Calculate the supplement angle.

  $\begin{array}{c}\pi -\frac{\pi }{3}=\frac{3\pi -\pi }{3}\\ =\frac{2\pi }{3}\end{array}$

Therefore, the complement and supplement of angle $\frac{\pi }{3}$ are $\frac{\pi }{6}$ and $\frac{2\pi }{3}$ , respectively.

b.

To determine

To find:The complement and supplement of angle $\frac{\pi }{4}$ .

Answer to Problem 18E

The complement and supplement of angle $\frac{\pi }{4}$ are $\frac{\pi }{4}$ and $\frac{3\pi }{4}$ , respectively.

Explanation of Solution

Given information:

The give angle is $\frac{\pi }{4}$ .

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.

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

Calculate the supplement angle.

  $\begin{array}{c}\pi -\frac{\pi }{4}=\frac{4\pi -\pi }{4}\\ =\frac{3\pi }{4}\end{array}$

Therefore, the complement and supplement of angle $\frac{\pi }{4}$ are $\frac{\pi }{4}$ and $\frac{3\pi }{4}$ , respectively.