Problem 78 - Precalculus

Question

Determine the angle of the smallest possible positive measure that is coterminal with each of the angles whose measure is given. Use degree or radian measures accordingly. $$-\frac{217 \pi}{4}$$

Step-by-Step Solution

Answer

The angle is \(\frac{3\pi}{4}\).

Step 1: Understanding Coterminal Angles

Coterminal angles share the same terminal side. To find them, we add or subtract full rotations (in radians, that's \(2\pi\)) from the given angle until arriving at a positive measure within one full rotation (\(0\) to \(2\pi\) radians).

Step 2: Full Rotation Recall

One complete circle in radians is \(2\pi\). Our goal is to add \(2\pi\) until the angle is between \(0\) and \(2\pi\).

Step 3: Adding Full Rotations

Add \(2\pi\) repeatedly to \(-\frac{217\pi}{4}\) until the angle falls within the range \(0\) to \(2\pi\). First, calculate \(2\pi\) in terms of fourths to make addition possible: \(2\pi = \frac{8\pi}{4}\).

Step 4: Performing the Additions

Start adding \(\frac{8\pi}{4}\) to \(-\frac{217\pi}{4}\):\(-\frac{217\pi}{4} + \frac{8\pi}{4} = -\frac{209\pi}{4}\). Continue this process as follows: - \(-\frac{209\pi}{4} + \frac{8\pi}{4} = -\frac{201\pi}{4}\)- \(-\frac{201\pi}{4} + \frac{8\pi}{4} = -\frac{193\pi}{4}\)- Repeat this step until you reach within the interval \(0\to\frac{8\pi}{4}\).

Step 5: Finding the Positive Coterminal Angle

Continue adding until the angle lies between \(0\) and \(2\pi\). After performing the appropriate number of additions (in this case 109 times), \(-\frac{217\pi}{4} + 109\times\frac{8\pi}{4} = \frac{3\pi}{4}\). The positive coterminal angle is therefore \(\frac{3\pi}{4}\).

Key Concepts

Radian Measure

The radian is a fundamental unit of angular measure used in many areas of mathematics, particularly in precalculus and calculus. It is based on the radius of a circle. One full circle equals an angle of

• \(2\pi\) radians.

This measure often simplifies the mathematics behind trigonometry and calculus. Unlike degrees, which divide a circle into 360 parts, radians offer a more natural way to understand angles through their relation to

• the arc length,
• where one radian is the angle created when the arc length is equal to the radius of the circle.

In this context, the exercise involves converting a negative angle in radians, \(-\frac{217\pi}{4}\), into a positive coterminal angle.

Full Rotation

A full rotation in a circle is the same as 360 degrees, or

• \(2\pi\) radians.

This is an important concept when dealing with coterminal angles, as it allows us to understand how an angle of any size can be related back to something more familiar within a full circle. By adding or subtracting full rotations from an angle, we

• 're-set' the angle to a measure within a single circle,
• from 0 to \(2\pi\) radians.

In the solution, we added multiples of

• \(2\pi\) until the negative angle opened positively within the complete range of a single circle.

Positive Angle

A positive angle is one that is measured in the counter-clockwise direction from the positive x-axis. In trigonometry and geometry, being able to quickly convert negative angles into positive is crucial for solving problems and comparing angles. To convert a negative angle into a coterminal positive angle, repeatedly add

• \(2\pi\) radians.

This process is reflected in our solution steps where we add

• \(\frac{8\pi}{4}\) multiple times until the angle was between \(0\) to \(2\pi\).

The result, in this practice, was a positive angle \(\frac{3\pi}{4}\), serving as the smallest positive coterminal angle for the original negative value.

Precalculus

Precalculus lays the foundation for understanding more advanced mathematics, specifically calculus. It encompasses several concepts including trigonometry, complex numbers, and mathematical analysis. A key aspect of precalculus is understanding the properties and calculations of angles in both degrees and radians, which are essential for building intuitive mathematical skills. This exercise exemplifies that by demanding students find coterminal angles, a common topic in this discipline. Utilizing these fundamental ideas prepares students for tackling topics such as

• derivatives, where angle measures and their properties play a crucial role,
• as well as integrals, which further employ these foundational concepts.

Understanding precalculus ensures students build a robust platform from which to delve into the world of calculus and beyond.