Problem 70 - Algebra and Trigonometry

Question

Find a positive angle less than \(360^{\circ}\) or \(2 \pi\) that is coterminal with the given angle. $$ -\frac{38 \pi}{9} $$

Step-by-Step Solution

Answer

The positive angle less than \(360^{\circ}\) or \(2 \pi\) that is coterminal with the given angle \(-\frac{38 \pi}{9}\) is \(\frac{16 \pi}{9}\).

Step 1: Identify the given angle

The given angle is \(-\frac{38 \pi}{9}\). Our aim is to find an equivalent positive angle less than \(2 \pi\).

Step 2: Convert the negative angle to a positive coterminal angle

This can be done by continuously adding \(2 \pi\) (a full circle) until the result falls within the desired range. Note that \(2 \pi\) can also be written as \(\frac{18 \pi}{9}\). So, add \(2 \pi\) or \(\frac{18 \pi}{9}\) to the given angle: \(-\frac{38 \pi}{9} + \frac{18 \pi}{9} = -\frac{20 \pi}{9}\). However, this is still not within the desired range. You need to repeat this process until you get an angle that is within the specified range.

Step 3: Repeat the process

So you repeat the process again. \(-\frac{20 \pi}{9} + \frac{18 \pi}{9} = -\frac{2 \pi}{9}\). This angle is still negative, so repeat the process once more. \(-\frac{2 \pi}{9} + \frac{18 \pi}{9} = \frac{16 \pi}{9}\). Now this is less than \(2 \pi\) and it is positive.

Key Concepts

Negative Angles

Negative angles might seem confusing at first, but they are simply rotations in the clockwise direction. When we deal with angles, usually the positive direction is counterclockwise. Therefore, a negative angle means spinning backwards from the standard position.

• Think of negative angles as moving the opposite way around a circle.
• If you start at 0° and rotate to -90°, you go clockwise to the 90° line.

For instance, the angle ewline\(-\frac{38 \pi}{9}\) represents such a backwards rotation. The concept is straightforward: just move in the reverse direction around the circle.
Once you know your angle is negative, to make calculations simpler, you can convert it to a positive angle. This brings us to understanding how positive angles work.

Positive Angles

Positive angles are more familiar and more commonly used. They represent a rotation in the counterclockwise direction from the starting line, which is typically the positive x-axis.

• Consider positive angles as moving normally around a circle, like following a clock moving forwards.
• A 360° or \(2\pi\) rotation brings you back to the start.

In mathematics, converting a negative angle to a positive one involves adding full circles (360° or \(2\pi\) radians).
Imagine you have an angle like ewline\(-\frac{38 \pi}{9}\) and you want a positive angle less than \(2\pi\). You just keep adding \(2\pi = \frac{18 \pi}{9}\) until your result fits within the range from 0 to \(2\pi\).
This means you're bringing the angle into a format that's easy to work with, using its positive equivalent.

Angle Conversion

Angle conversion is about changing an angle from one form to another, typically between degrees and radians, or converting negative angles into positive angles within a certain range.

Converting Between Degrees and Radians

To switch between degrees and radians, use the basic relationship: \(180^{\circ} = \pi\) radians.

• To convert degrees to radians, multiply the degree value by \(\frac{\pi}{180}\).
• To convert radians to degrees, multiply the radian value by \(\frac{180}{\pi}\).

Finding Coterminal Angles

Coterminal angles share the same initial and terminal sides. You can find them by adding or subtracting a full circle (360° or \(2\pi\) radians):

• For the angle \(-\frac{38 \pi}{9}\), adding \(\frac{18 \pi}{9}\) repeatedly helps find its positive coterminal angle as described in the solution.
• This approach ensures your angle is properly formatted for specific applications.

Understanding and performing these conversions allows you to handle angles effectively, making mathematical problems easier to explore and solve.