Problem 31 - Precalculus: A Concise Course

Question

Find (if possible) the complement and supplement of each angle. (a) \(\pi / 3\) (b) \(\pi / 4\)

Step-by-Step Solution

Answer

The complement of \(\frac{\pi}{3}\) is \(\frac{\pi}{6}\) and its supplement is \(\frac{2\pi}{3}\). The complement of \(\frac{\pi}{4}\) is the same angle itself, \(\frac{\pi}{4}\), while its supplement is \(\frac{3\pi}{4}\).

Step 1: Finding the Complement and Supplement of \(\frac{\pi}{3}\)

To find the complement, subtract the given angle from \(\frac{\pi}{2}\). So, \(\frac{\pi}{2} - \frac{\pi}{3} = \frac{\pi}{6}\). To find the supplement, subtract the given angle from \(\pi), yielding \(\pi - \frac{\pi}{3} = \frac{2\pi}{3}\).

Step 2: Finding the Complement and Supplement of \(\frac{\pi}{4}\)

Meanwhile, the complement of \(\frac{\pi}{4}\) can be calculated by \(\frac{\pi}{2} - \frac{\pi}{4} = \frac{\pi}{4}\), and the supplement can be found by subtracting the given angle from \(\pi), arriving at \(\pi - \frac{\pi}{4} = \frac{3\pi}{4}\).

Key Concepts

Radians

Radians are a fundamental unit of angle measurement used extensively in mathematics, especially in trigonometry. Unlike degrees, which divide a circle into 360 parts, radians relate directly to the radius of a circle. This provides a more natural and convenient way to express angles, especially in mathematical formulas.

One radian is defined as the angle created when the arc length is equal to the radius of the circle. Therefore, a full circle, which is the full circumference divided by the radius, is equivalent to:

• 2π radians, as the circumference of a circle is \(2\pi r\).

Common angles, such as \(\pi/3\) and \(\pi/4\), denote specific portions of the circle and are often used in calculations involving trigonometric functions and periodic phenomena.

Working with radians is essential when dealing with trigonometric identities, calculus, and many physics applications, where angles directly influence rates of change and vector directions.

Trigonometry

Trigonometry is the branch of mathematics dealing with angles, triangles, and the relationships between their sides and angles. At its core, trigonometry is all about finding how these parts relate to each other within right triangles, and these concepts are extrapolated to non-right triangles using laws like the Law of Sines and Cosines.

Trigonometry involves several key functions:

• Sine (sin),
• Cosine (cos),
• Tangent (tan).

These functions are essential in calculating angles and distances in fields such as physics, engineering, and astronomy.

Complementary and supplementary angles are crucial concepts in trigonometry:

• Complementary angles add up to \(\frac{\pi}{2}\) radians (or 90°).
• Supplementary angles add up to \(\pi\) radians (or 180°).

Understanding these relationships helps in solving trigonometric equations and analyzing geometric properties.

Angle Calculation

Calculating angles, specifically their complements and supplements, is a fundamental skill in trigonometry and geometry. The complement of an angle is what, when added to the original angle, results in \(\frac{\pi}{2}\) radians or 90 degrees.

For example, to find the complement of an angle like \(\pi/3\):

• Subtract it from \(\frac{\pi}{2}\).

This means \(\frac{\pi}{2} - \frac{\pi}{3} = \frac{\pi}{6}\).

Similarly, the supplement is determined by subtracting the angle from \(\pi\). For instance:

• For an angle of \(\pi/4\), its supplement is \(\pi - \frac{\pi}{4} = \frac{3\pi}{4}\).

These calculations help us understand and determine various geometric configurations and are indispensable in fields that depend on precise angle measurements, such as architecture and navigation.