Problem 49 - Precalculus Mathematics for Calculus

Question

Find an angle between 0 and \(2 \pi\) that is coterminal with the given angle. $$\frac{17 \pi}{4}$$

Step-by-Step Solution

Answer

The angle coterminal with \( \frac{17 \pi}{4} \) between 0 and \( 2\pi \) is \( \frac{\pi}{4} \).

Step 1: Convert Angle to Common Denominator

To solve this, we need to find an angle coterminal with \( \frac{17 \pi}{4} \) that lies between 0 and \( 2\pi \). First, we convert \( 2\pi \) to a fraction with the same denominator as \( \frac{17 \pi}{4} \). We have \( 2\pi = \frac{8\pi}{4} \).

Step 2: Subtract Multiples of \(2\pi\)

Subtract \( 2\pi \) from \( \frac{17 \pi}{4} \):\[\frac{17 \pi}{4} - \frac{8 \pi}{4} = \frac{9 \pi}{4}\]This angle \( \frac{9 \pi}{4} \) is still greater than \( 2\pi \), repeat subtracting.

Step 3: Subtract Again to Find Coterminal Angle

Subtract \( 2\pi \) once more from \( \frac{9 \pi}{4} \):\[\frac{9 \pi}{4} - \frac{8 \pi}{4} = \frac{\pi}{4}\]Now the result \( \frac{\pi}{4} \) is between 0 and \( 2\pi \) and is the coterminal angle.

Key Concepts

Angle Conversion

Understanding angle conversion is essential when working with trigonometry problems like finding coterminal angles. Angles can be expressed in various units, typically degrees or radians. In this case, we are dealing with radians, which is the standard unit in the context of trigonometry and calculus. One full revolution around a circle is equivalent to \(2\pi\) radians.

To work with fractions, it's often beneficial to convert values to a common denominator to streamline calculations. For instance, converting \(2\pi\) to \(\frac{8\pi}{4}\) allows us to easily compare and perform operations with \(\frac{17\pi}{4}\). This common denominator simplifies subtraction and subsequent steps in solving for coterminal angles. By consistently applying this conversion technique, you ensure accuracy in your trigonometric calculations.

Angle Subtraction

In finding coterminal angles, angle subtraction becomes key. Coterminal angles are distinct angles that share a terminal side when plotted in standard position. To identify a coterminal angle between 0 and \(2\pi\), we subtract multiples of \(2\pi\) from the given angle until it falls within the desired range.

Begin by subtracting \(\frac{8\pi}{4}\) from \(\frac{17\pi}{4}\), which gives us \(\frac{9\pi}{4}\). Since \(\frac{9\pi}{4}\) remains greater than \(2\pi\), repeat the subtraction. The second subtraction results in \(\frac{\pi}{4}\), which fits within our specified range (0 to \(2\pi\)). This iterative subtraction process is fundamental when refining angles to identify their coterminal counterparts.

Trigonometry

Trigonometry revolves around studying relationships involving angles and triangles. Coterminal angles are a crucial concept within this field. They demonstrate how different angle measures can relate through rotations around the unit circle.

When solving problems that require determining coterminal angles, trigonometry provides tools and techniques to simplify these tasks. The unit circle, measuring angles in radians, allows us to see how angles greater than \(2\pi\) or less than 0 can be "wrapped" back into a fundamental circle (0 to \(2\pi\)). Understanding these foundational trigonometric concepts is invaluable for advanced studies in mathematics, physics, and engineering, where angle measures frequently recur and interconnect.