Question

Find the reference angle of each angle. $$ -\frac{5 \pi}{7} $$

Short answer

\( \frac{2\pi}{7} \)

Step-by-step solution

Understand Reference Angle

The reference angle is the smallest angle between the given angle and the x-axis. It's always a positive acute angle (between 0 and \( \frac{\pi}{2} \)).

Determine the Equivalent Positive Angle

If an angle is negative, to find its positive equivalent, add \( 2\pi \) until the angle is positive. \(-\frac{5\pi}{7} + 2\pi = -\frac{5\pi}{7} + \frac{14\pi}{7} = \frac{9\pi}{7} \).

Find the Reference Angle

Determine the reference angle when \( \frac{9\pi}{7} \) is greater than \( \pi \). Since \( \frac{9\pi}{7} \) is more than \( \pi (\frac{7\pi}{7}) \), the reference angle is \(-\pi + \frac{9\pi}{7} = \frac{2\pi}{7} \).

Key concepts

Understanding Negative Angles

A negative angle is one that is measured in the clockwise direction from the positive x-axis. By convention, angles are usually measured positively, in a counterclockwise direction. To convert a negative angle into an easier-to-work-with positive angle, we can keep adding full rotations of \( 2\pi \) (360 degrees) until the angle becomes positive. This method keeps the geometric properties intact and makes calculations easier. For example, in our problem, \( -\frac{5\pi}{7} \) is a negative angle. To find its positive equivalent, we add \( 2\pi \). This transforms it into \( \frac{9\pi}{7} \), a positive angle within the same geometric context.

Finding the Positive Equivalent

To find the positive equivalent of a negative angle, simply add \( 2\pi \) to the angle as many times as needed until you get a positive result. Here's a step-by-step guide:

• First, identify the negative angle. In this example, it's \( -\frac{5\pi}{7} \).
• Next, add \( 2\pi \) to this angle: \( -\frac{5\pi}{7} + 2\pi \).
• Convert \( 2\pi \) to a fraction with the same denominator for easier addition: \( 2\pi = \frac{14\pi}{7} \).
• Add the fractions: \( -\frac{5\pi}{7} + \frac{14\pi}{7} = \frac{9\pi}{7} \).

You now have \( \frac{9\pi}{7} \), the positive equivalent of the original negative angle.
This equivalent angle retains the same geometric properties while being easier to work with.

Reference Angle and Acute Angles

The reference angle is the smallest angle between the given angle and the x-axis. It's always a positive acute angle, meaning it lies between 0 and \( \frac{\pi}{2} \) (0 and 90 degrees). To find the reference angle:

• Check if the angle is more significant than \( \pi \) (180 degrees). If it is, subtract \( \pi \) from it.
• For \( \frac{9\pi}{7} \), since this angle is greater than \( \pi \), subtract \( \pi \): \( \frac{9\pi}{7} - \pi = \frac{2\pi}{7} \).

The result, \( \frac{2\pi}{7} \), is your reference angle. Notice it's positive and less than \( \frac{\pi}{2} \), making it an acute angle.
Understanding reference angles helps simplify problems in trigonometry, making other calculations involving sine, cosine, and tangent easier.