Problem 57 - Precalculus

Question

For the following exercises, find the between 0 and 2\(\pi\) in radians that is coterminal to the given angle. $$ \frac{44 \pi}{9} $$

Step-by-Step Solution

Answer

The coterminal angle is \(\frac{8\pi}{9}\).

Step 1: Understanding Coterminal Angles

Coterminal angles are angles that share the same terminal side. To find a coterminal angle between 0 and \(2\pi\), we add or subtract multiples of \(2\pi\) until the result falls within this range.

Step 2: Convert Angle to Coterminal Range

Start by subtracting \(2\pi\) (since \(2\pi = \frac{18\pi}{9}\)) from \(\frac{44\pi}{9}\) to bring the angle closer to the desired range. Calculate: \(\frac{44\pi}{9} - \frac{18\pi}{9} = \frac{26\pi}{9}\).

Step 3: Subtract \(2\pi\) Again

Since \(\frac{26\pi}{9}\) is still greater than \(2\pi\), subtract \(2\pi\) once more: \(\frac{26\pi}{9} - \frac{18\pi}{9} = \frac{8\pi}{9}\).

Step 4: Check Coterminal Range

Verify if \(\frac{8\pi}{9}\) is between 0 and \(2\pi\). Since \(\frac{8\pi}{9} \approx 2.79\) is less than \(2\pi \approx 6.28\), it fits within the range.

Key Concepts

Radian Measure

To understand coterminal angles effectively, it's essential to grasp the concept of radian measure. Radians provide an alternative way to measure angles, compared to degrees. Here's the basic idea:

• A radian is defined as the angle created when the radius of a circle is wrapped along its circumference.
• In one full circle, there are \(2\pi\) radians, equivalent to 360 degrees.

This makes the conversion between degrees and radians straightforward:

• Degrees to Radians: Multiply by \(\frac{\pi}{180}\)
• Radians to Degrees: Multiply by \(\frac{180}{\pi}\)

Using radians often makes calculations, especially in calculus and trigonometry, more straightforward and elegant. When working with problems on coterminal angles, always ensure your angles are in radian measure for consistency.

Angle Subtraction

When we talk about angle subtraction in the context of finding coterminal angles, the objective is to adjust your given angle so it fits within a particular range.For this problem, you have the angle \(\frac{44\pi}{9}\) in radians. Here's how to manage it:

• Coterminal angles mean we want angles that might start from different rotations, but end up with the same terminal side.
• The rotation for one full circle is \(2\pi\), which can be added or subtracted from any angle.

To adjust your angle, subtract multiples of \(2\pi\) until your angle is within the range 0 to \(2\pi\). In this exercise:

• Subtract \(\frac{18\pi}{9}\) (which is \(2\pi\)) to decrease \(\frac{44\pi}{9}\) to \(\frac{26\pi}{9}\).
• Subtract again to make it \(\frac{8\pi}{9}\), now comfortably within 0 to \(2\pi\).

Angle Range

The angle range concept is crucial when determining coterminal angles. We want our angles to be between 0 and \(2\pi\) radians.Here's why this range matters:

• Angles between 0 and \(2\pi\) reside in a single revolution on a circle's circumference.
• We typically express angles within this range for simplicity and consistency.

With the original exercise, the task was to ensure that the final angle, \(\frac{8\pi}{9}\), sits within the optimal interval.

• Since \(\frac{8\pi}{9}\) is approximately 2.79 in a linear numerical sense, it is less than \(2\pi\) (about 6.28), thus it qualifies.

Always double-check your adjustments to see if they fit within this range to verify your answer is correct and coterminal.