Problem 12 - A Graphical Approach to Precalculus with Limits

Question

Find (a) the complement and (b) the supplement of each angle. Do not use a calculator. $$\frac{\pi}{12}$$

Step-by-Step Solution

Answer

Complement: \( \frac{5\pi}{12} \), Supplement: \( \frac{11\pi}{12} \)

Step 1: Understanding the Problem

We are given an angle of \( \frac{\pi}{12} \) radians and asked to find its complement and supplement. The complement of an angle in radians is found by subtracting the angle from \( \frac{\pi}{2} \), and the supplement is found by subtracting the angle from \( \pi \).

Step 2: Finding the Complement

To find the complement of \( \frac{\pi}{12} \), subtract it from \( \frac{\pi}{2} \).\[ \text{Complement} = \frac{\pi}{2} - \frac{\pi}{12} \]}{

Step 3: Calculating the Complement

First, express \( \frac{\pi}{2} \) in terms of \( \frac{\pi}{12} \) to simplify the subtraction.\[ \frac{\pi}{2} = \frac{6\pi}{12} \]Now subtract \( \frac{\pi}{12} \) from \( \frac{6\pi}{12} \):\[ \frac{6\pi}{12} - \frac{\pi}{12} = \frac{5\pi}{12} \]Thus, the complement is \( \frac{5\pi}{12} \).

Step 4: Finding the Supplement

To find the supplement of \( \frac{\pi}{12} \), subtract it from \( \pi \).\[ \text{Supplement} = \pi - \frac{\pi}{12} \]}{

Step 5: Calculating the Supplement

Express \( \pi \) in terms of \( \frac{\pi}{12} \) to simplify the subtraction.\[ \pi = \frac{12\pi}{12} \]Now subtract \( \frac{\pi}{12} \) from \( \frac{12\pi}{12} \): \[ \frac{12\pi}{12} - \frac{\pi}{12} = \frac{11\pi}{12} \]Thus, the supplement is \( \frac{11\pi}{12} \).

Key Concepts

Radians

Radians are units used to measure angles, similar to degrees but based on the properties of circles. Unlike degrees, which divide a circle into 360 parts, radians work with the circle’s radius. One full circle in radians is equal to \(2\pi\), which is approximately 6.28.

Here's an easy way to understand this: if you take the radius of a circle and "wrap" it along the circle's edge, the length will fit \(2\pi\) times around the circle. This relationship makes radians a natural way to express angles directly tied to the circle's arc length. Radians are especially useful in higher mathematics like calculus, where they simplify formulas and calculations.

Angle Complement

The complement of an angle refers to what we need to add to that angle to make it \(90^\circ\) or \(\frac{\pi}{2}\) radians, a right angle. In trigonometry, this is a commonly used concept. For instance, the sum of the complements of two angles is \(\frac{\pi}{2}\).

To find the complement of an angle in radians, subtract it from \(\frac{\pi}{2}\). As shown in the exercise, for an angle of \(\frac{\pi}{12}\), the calculation is:

• Complement = \(\frac{\pi}{2} - \frac{\pi}{12}\)
• Express \(\frac{\pi}{2}\) as \(\frac{6\pi}{12}\)
• Subtract to find the complement: \(\frac{6\pi}{12} - \frac{\pi}{12} = \frac{5\pi}{12}\)

The angle complement helps in various geometrical problems and trigonometric identities.

Angle Supplement

The supplement of an angle is what you add to it to reach a straight angle, \(180^\circ\) or \(\pi\) radians. If two angles are supplementary, their sum equals \(\pi\). This concept helps in solving equilibrium problems in physics, among others.

For instance, to find the supplement of an angle of \(\frac{\pi}{12}\), the procedure is:

• Supplement = \(\pi - \frac{\pi}{12}\)
• Express \(\pi\) as \(\frac{12\pi}{12}\)
• Subtract to obtain \(\frac{12\pi}{12} - \frac{\pi}{12} = \frac{11\pi}{12}\)

This process shows how angles complement and supplement each other to equate to foundational benchmarks like a right angle and a straight angle.

Mathematical Subtraction

Mathematical subtraction is a fundamental operation used to find the difference between two values. In problems involving angles, subtraction helps determine what is needed to reach a certain total, such as a right angle or a straight angle.

In the context of finding complements and supplements, subtraction is key. Subtracting \(\frac{\pi}{12}\) from \(\frac{\pi}{2}\) or \(\pi\) helps us understand how much more is needed to reach those specific angles.

• In complement calculation: \(\frac{6\pi}{12} - \frac{\pi}{12} = \frac{5\pi}{12}\)
• In supplement calculation: \(\frac{12\pi}{12} - \frac{\pi}{12} = \frac{11\pi}{12}\)

Understanding this operation deeply helps not only in geometry but also in a wide range of mathematical fields.