Problem 45 - Algebra and Trigonometry

Question

Find an angle between 0 and 2p that is coterminal with the given angle. $$ \frac{17 \pi}{6} $$

Step-by-Step Solution

Answer

\(\frac{5\pi}{6}\) is coterminal with \(\frac{17\pi}{6}\).

Step 1: Understand Coterminal Angles

Two angles are coterminal if they share the same initial and terminal sides, which can be achieved by adding or subtracting full rotations (in radians, this is multiples of \(2\pi\)). We want an angle between 0 and \(2\pi\) that is coterminal with \(\frac{17\pi}{6}\).

Step 2: Convert Angle to a Standard Form

To find a coterminal angle between 0 and \(2\pi\), subtract \(2\pi\) from \(\frac{17\pi}{6}\) because \(\frac{17\pi}{6}\) is greater than \(2\pi\).

Step 3: Perform the Calculation

Calculate \(\frac{17\pi}{6} - 2\pi\). First, express \(2\pi\) with a common denominator of 6: \(2\pi = \frac{12\pi}{6}\). Then: \[ \frac{17\pi}{6} - \frac{12\pi}{6} = \frac{5\pi}{6} \]

Step 4: Verify the Result

The result \(\frac{5\pi}{6}\) is between 0 and \(2\pi\), making it a valid coterminal angle. It is indeed one rotation of \(2\pi\) less than \(\frac{17\pi}{6}\), thus confirming its coterminality.

Key Concepts

Radian Measure

When we talk about angles in trigonometry, they are often measured in radians. A radian is one way to express the magnitude of an angle based on the radius of a circle. While degrees are more commonly used, radians offer a more natural mathematical context for calculus and trigonometry because they are directly related to the circle's geometry.

To better understand, consider a full circle:

• A full circle is equivalent to an angular measure of 360 degrees or \(2\pi\) radians.
• Half a circle, or a semicircle, measures 180 degrees or \(\pi\) radians.
• When calculating or converting angles, recognizing these fundamental equivalents can make the process easier.

In the context of this exercise, \(\frac{17\pi}{6}\) represents an angle in radians, which we aim to convert into one that falls between 0 and \(2\pi\) radians to find a coterminal angle.

Angle Conversion

Angle conversion is the practice of switching between different measurement systems or bringing measurements into a simpler form. In this exercise, we are not converting between degrees and radians but rather simplifying the current radian measurement to an equivalent angle that falls within a convenient range (0 to \(2\pi\)).

To convert angles, we often add or subtract full rotations ( \(2\pi\) radians) until the angle lies within the desired range. Here’s how you can approach such conversions:

• Start by identifying if your original angle is larger or smaller than the target range.
• Subtract increments of \(2\pi\) (for reducing) or add \(2\pi\) (for increasing) until the angle falls within 0 and \(2\pi\).
• Expressing \(2\pi\) with a suitable denominator helps in performing the arithmetic. For \(\frac{17\pi}{6}\), it was expressed as \(\frac{12\pi}{6}\) to maintain a common denominator for subtraction.

This technique helps find an angle that looks simpler and is more practical to interpret in the context of a circle.

Trigonometry

Trigonometry studies relationships between side lengths and angles in triangles, but its principles extend further, including the study of angles on a unit circle. Understanding coterminal angles is a key topic in this field, as these angles represent the same trigonometric functions (like sine or cosine) because they have the same terminal points on the circle.

When two angles are coterminal, their trigonometric values like \(\sin\), \(\cos\), and \(\tan\) are equivalent, which is practical for solving many problems involving periodic functions. Here’s why coterminal angles are significant:

• They help in simplifying calculations as, for practical purposes, a \(360^\circ\) or \(2\pi\) shift doesn't change the result.
• Working with smaller, equivalent angles is often easier and allows one to focus on key properties without dealing with excessively large values.
• This concept is crucial in analyzing wave patterns, harmonic motions, and engineering applications where angles repeat over cycles.

By understanding this relationship, mastering trigonometry becomes both simpler and more insightful, unlocking a deeper comprehension of both theoretical and applied mathematics.