Problem 92 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find (if possible) the complement and supplement of the angle. $$\frac{12 \pi}{5}$$

Step-by-Step Solution

Answer

The angle does not have a complement or a supplement.

Step 1: Convert to Degrees

Firstly, the given angle which is in radians must be converted to degrees using the formula : \(\frac{12 \pi}{5} × \frac{180}{\pi}\) which gives around 216.818 degrees.

Step 2: Complement of the Angle

Normally, the complement of the angle is calculated by subtracting the angle from 90 degrees. However, as our angle is greater than 90 degrees, it cannot have a complement. Hence, the complement of the angle does not exist.

Step 3: Supplement of the Angle

The supplement of the angle is calculated by subtracting the angle from 180 degrees: \(180 - 216.818 = -36.818 \) degrees. As the result is negative, the angle does not have a supplement.

Key Concepts

Complementary Angles

Complementary angles are a pair of angles whose sum is exactly 90 degrees. When angles are complementary, they fit together to form a right angle. For example, if you have one angle that measures 30 degrees, its complement must measure 60 degrees to add up to 90 degrees together.

However, not every angle can have a complement. For instance, if an angle measures more than 90 degrees, it is already larger than a right angle on its own. Thus, it can't be paired with another angle to achieve a total of 90 degrees.

In the original exercise, after converting from radians to degrees, the angle is about 216.818 degrees. As this angle exceeds 90 degrees, it inherently cannot have a complementary angle. This concept helps in understanding the limitation when calculating complementary angles.

Supplementary Angles

Supplementary angles are two angles whose measures add up to 180 degrees. These angles can be thought of as forming a straight line when placed adjacent to each other. For example, a 110-degree angle and a 70-degree angle are supplementary because they sum to 180 degrees.

However, similar to complementary angles, not all angles can have a supplementary counterpart. If an angle exceeds 180 degrees, it cannot form a straight line with another angle through addition.

In the exercise, the given angle measures about 216.818 degrees after converting from radians to degrees. Attempting to find its supplementary angle by subtracting from 180 degrees ( 180 - 216.818 ) results in a negative angle measurement, -36.818 degrees, indicating no valid supplementary angle. This scenario explains how angles larger than 180 degrees do not have supplements, reinforcing the rules of supplementary angles.

Radian to Degree Conversion

Radian and degree are two units to measure angles. Conversion between them is crucial, particularly when working with angles in different contexts such as trigonometry or physics.

The formula for converting an angle from radians to degrees is given by: \(\theta_{degrees} = \theta_{radians} \times \frac{180}{\pi}\). The factor \(\frac{180}{\pi}\) converts the angle measured in radians into degrees. This conversion is necessary as most people and even some fields prefer degrees when discussing angles.

In the original exercise, the angle \(\frac{12 \pi}{5}\) radians was converted to degrees as follows:\(\frac{12 \pi}{5} \times \frac{180}{\pi} = 216.818\) degrees. Understanding this conversion process enables you to seamlessly move between radian and degree measures, which is essential in mathematical and scientific computations.