Problem 120 - Algebra and Trigonometry

Question

Determine whether each statement makes sense or does not make sense, and explain your reasoning. Using radian measure, I can always find a positive angle less than \(2 \pi\) coterminal with a given angle by adding or subtracting \(2 \pi\)

Step-by-Step Solution

Answer

The statement makes sense because by adding or subtracting multiples of \(2 \pi \) from an angle in radian measure, a positive coterminal angle less than \(2 \pi \) can always be found.

Step 1: Understanding Co-terminal Angles

Two angles are coterminal if they have the same terminal side. This also means that one angle can be obtained by adding or subtracting multiples of \(2 \pi \) (a complete revolution) to the other angle. In other words, any angle \(x\) and \(x + 2 \pi n\) (where \(n\) is an integer) are coterminal.

Step 2: Validating the Statement

The statement given says that by using radian measure, a positive angle, less than \(2 \pi \), that's coterminal with a given angle can always be found by adding or subtracting \(2 \pi \). This statement is true. For a given angle, if the angle measure is greater than \(2 \pi \), subtract \(2 \pi \) from the angle to get a positive angle less than \(2 \pi \). If the angle measure is negative, add \(2 \pi \) to the angle to get a positive angle less than \(2 \pi \).

Step 3: Explaining the Reasoning

This concept depends on the cyclic nature of angles and the fact that adding or subtracting full rotations does not change the terminal position of the angle, hence the concept of coterminal angles. By adding or subtracting multiples of \(2 \pi \), the angle can be adjusted to a desired range while preserving its terminal side location and therefore coterminality. For instance, if the given angle is \(5 \pi \), subtracting \(2 \pi \) (twice) gives \( \pi \) which is less than \(2 \pi \) and coterminal with original angle \(5 \pi \).