Problem 47 - Algebra 2

Question

Find one angle with positive measure and one angle with negative measure coterminal with each angle. $$ \frac{5 \pi}{3} $$

Step-by-Step Solution

Answer

Positive: \(\frac{11\pi}{3}\); Negative: \(-\frac{\pi}{3}\).

Step 1: Understand Coterminal Angles

Coterminal angles are angles that, when sketched in standard position (with the initial side on the positive x-axis), share the same terminal side. To find angles coterminal with a given angle, we add or subtract multiples of a full circle. In radians, a full circle is \[ 2\pi. \]

Step 2: Find a Positive Coterminal Angle

To find a positive coterminal angle, add \(2\pi\) to the given angle. The given angle is \(\frac{5\pi}{3}\), thus: \[ \frac{5\pi}{3} + 2\pi = \frac{5\pi}{3} + \frac{6\pi}{3} = \frac{11\pi}{3}. \] This means \(\frac{11\pi}{3}\) is one positive angle coterminal with \(\frac{5\pi}{3}.\)

Step 3: Find a Negative Coterminal Angle

To find a negative coterminal angle, subtract \(2\pi\) from the given angle. Starting with \(\frac{5\pi}{3}\), we have: \[ \frac{5\pi}{3} - 2\pi = \frac{5\pi}{3} - \frac{6\pi}{3} = -\frac{\pi}{3}. \] This means \(-\frac{\pi}{3}\) is one negative angle coterminal with \(\frac{5\pi}{3}.\)

Key Concepts

Understanding Angles

Angles are geometric figures formed by two rays, known as sides, that share a common endpoint called the vertex. They are pivotal in geometry and trigonometry, describing rotational positions and measures. Angles can be measured in several ways, with degrees and radians being the most common.

In degree measurement, a full circle is divided into 360 parts, known as degrees. This division stems from ancient astronomical calculations. Alternatively, radians are the building blocks used in more advanced mathematics, providing an intrinsic measurement connected to the properties of a circle itself.

It's essential not only to understand what angles are but also to visualize them in different scenarios. This ability to visualize is crucial in identifying patterns such as coterminal angles, which we explore next.

Radians as a Measure

Radians are a unit of angular measure based on the radius of a circle. Unlike degrees, which divide a circle into arbitrary parts, radians relate more directly to the circle's geometry.

A radian measures an angle created by wrapping the radius of a circle along its circumference. Because of this, a complete circle encompasses \(2\pi\) radians. This relationship allows for more natural integration into calculus and higher mathematical concepts, making radians a preferred unit in many fields.

When converting between radians and degrees, use the relationship \( \pi \text{ radians} = 180^\circ \). This conversion method helps switch between systems. Understanding radians is crucial as they frequently surface in trigonometric functions, calculus, and when determining coterminal angles.

Coterminal Angle Computations

Coterminal angles are angles that share a terminal side when drawn in standard position. They are separated by full circle turns, either positive or negative.

To find coterminal angles for a given radian, you need to either add or subtract multiples of \( 2\pi \). This is because \( 2\pi \) represents a full rotation around the circle. For example, if you have the angle \( \frac{5\pi}{3} \), adding \( 2\pi \) gives \( \frac{11\pi}{3} \) as a positive coterminal angle. Similarly, subtracting \( 2\pi \) results in \(-\frac{\pi}{3} \), a negative coterminal angle.

This process highlights how coterminal angles differ only by complete rotations. Mastery of this concept supports solving many trigonometry problems and enhances understanding of circular motion, wave functions, and periodic events.