Problem 32 - Precalculus: A Concise Course

Question

Find (if possible) the complement and supplement of each angle. (a) \(\pi / 12\) (b) \(11 \pi / 12\)

Step-by-Step Solution

Answer

The complement of \(\pi/12\) is \(\pi/2 - \(\pi/12 = \(5\pi/12\) and the supplement is \(\pi - \(\pi/12 = \(11\pi/12\). The complement of \(11\pi/12\) doesn't exist, as the initial angle is greater than \(\pi/2\). The supplement of \(11\pi/12\) is \(\pi - 11\pi/12 = \pi/12\).

Step 1: Find complements

Calculate the complement angles of both \(\pi/12\) and \(11\pi/12\) by subtracting them from \(\pi/2\).

Step 2: Find supplements

Subtract the angles \(\pi/12\) and \(11\pi/12\) from \(\pi\) to get the supplement angles.

Key Concepts

Complementary Angles

Complementary angles are two angles whose measures add up to 90 degrees. In the realm of radians, this translates to the sum of the two angles being equal to \( \frac{\pi}{2} \). This means if you have one angle, its complement is found by subtracting it from \( \frac{\pi}{2} \).

For example, if we have an angle of \( \frac{\pi}{12} \), its complementary angle would be calculated as follows:
\[ \text{Complement of } \frac{\pi}{12} = \frac{\pi}{2} - \frac{\pi}{12} = \frac{6\pi}{12} - \frac{\pi}{12} = \frac{5\pi}{12} \]

Always remember, not all angles have a complement, especially if the given angle is already greater than \( \frac{\pi}{2} \), like \( \frac{11\pi}{12} \). In such cases, the complement does not exist because it would have to be negative to add up to \( \frac{\pi}{2} \).

Complementary angles often appear in right-angled triangles, where two non-right angles must always add up to 90 degrees or \( \frac{\pi}{2} \). This is very useful in trigonometry and geometry.

Supplementary Angles

Supplementary angles are pairs of angles whose measures sum up to 180 degrees or simply \( \pi \) radians. To find the supplement of an angle, subtract it from \( \pi \).

Suppose we need to find the supplement of \( \frac{\pi}{12} \). You would do this subtraction:
\[ \text{Supplement of } \frac{\pi}{12} = \pi - \frac{\pi}{12} = \frac{12\pi}{12} - \frac{\pi}{12} = \frac{11\pi}{12} \]

Similarly, the supplement of \( \frac{11\pi}{12} \) is calculated as follows:
\[ \text{Supplement of } \frac{11\pi}{12} = \pi - \frac{11\pi}{12} = \frac{12\pi}{12} - \frac{11\pi}{12} = \frac{\pi}{12} \]

Note that unlike complementary angles, all angles have a supplement, since any angle subtracted from \( \pi \) will result in another valid angle measure. Supplementary angles often feature in linear pairs and straight lines in geometry, thus playing a crucial role in defining shapes and proving theorems.

Radians

Radians are an alternative to degrees for measuring angles, and often used in advanced mathematics. One radian is the angle created when the arc length on a circle is equal to the circle's radius. This means that a full circle, which has an arc length equal to the circumference, measures \( 2\pi \) radians.

So let’s break it down:

• \( \pi \) radians correspond to half a full circle, or 180 degrees.
• \( \frac{\pi}{2} \) radians equal a quarter of a circle, or 90 degrees.
• \( \frac{\pi}{12} \) radians is equal to dividing 180 degrees by 12, which results in 15 degrees.

Understanding radians helps with performing calculations in calculus, physics, and other sciences because many formulas use this unit of measure.

In trigonometry, converting between degrees and radians can be necessary depending on the given problem or function. Remember this conversion:
\[ 180^{\circ} = \pi \, \text{radians} \]

Radians simplify many mathematical expressions and are widely used in mathematical fields dealing with periodic functions.

Angle Subtraction

Angle subtraction is a crucial concept when dealing with both degrees and radians. Subtracting angles is often used to determine complementary and supplementary angles, as seen in trigonometry problems.

To recap from our previous sections:

• For complementary angles, subtract the given angle from \( \frac{\pi}{2} \) radians.
• For supplementary angles, subtract from \( \pi \) radians.

Performing the actual subtraction involves ensuring the angles are in the same measurement unit (radians in this context) and simplifying by finding a common denominator.

Let's see an example using \( \frac{\pi}{12} \):

• To find the complement: \( \frac{\pi}{2} - \frac{\pi}{12} = \frac{6\pi}{12} - \frac{\pi}{12} = \frac{5\pi}{12} \)
• To find the supplement: \( \pi - \frac{\pi}{12} = \frac{12\pi}{12} - \frac{\pi}{12} = \frac{11\pi}{12} \)

This simple arithmetic makes angle subtraction an easy-to-use tool in various mathematical problems and scenarios.