Question

Find the reference angle of each angle. $$ -\frac{7 \pi}{6} $$

Short answer

The reference angle is \( \frac{\text{π}}{6} \).

Step-by-step solution

Understand the Concept of a Reference Angle

A reference angle is the positive acute angle between the terminal side of the given angle and the x-axis. It is always between 0 and \(\frac{\text{π}}{2}\) radians.

Convert the Negative Angle to a Positive Angle

Add 2π to the given angle \(-\frac{7\text{π}}{6}\) since an angle plus 2π is coterminal with the original angle and will lie in the same position. \(-\frac{7\text{π}}{6} + 2\text{π} = \ -\frac{7\text{π}}{6} + \frac{12\text{π}}{6} = \frac{5\text{π}}{6} \)

Identify the Quadrant of the Angle

Since \(\frac{5\text{π}}{6}\) is in the second quadrant (because it is between \(\frac{\text{π}}{2}\) and \(\text{π}\)), the reference angle is \(\text{π}\) minus the angle.

Calculate the Reference Angle

Subtract the given angle from π to find the reference angle: \(\text{π} - \frac{5\text{π}}{6} = \frac{6\text{π}}{6} - \frac{5\text{π}}{6} = \frac{\text{π}}{6} \)

Key concepts

Negative Angles

Understanding how to work with negative angles is crucial in trigonometry. A negative angle is one that is measured in the clockwise direction from the positive x-axis.

For example, in the exercise given, \(-\frac{7\pi}{6}\) is a negative angle.

To find its reference angle, we first need to convert this negative angle to a positive one by adding 2\(\pi\). This is because an angle plus 2\(\pi\) (or 360°) is coterminal with the original angle, meaning they point in the same direction.

So, if we add 2\pi to -\frac{7\pi}{6}, we get: \-\frac{7\pi}{6} + 2\pi = -\frac{7\pi}{6} + \frac{12\pi}{6} = \frac{5\pi}{6}\

Coterminal Angles

Coterminal angles are angles that share the same initial and terminal sides but may have different measures. Understanding coterminal angles helps in simplifying problems, especially when working with negative angles or finding reference angles.

Two angles are coterminal if they differ by a multiple of 360° or 2\pi radians. In this context, to make a negative angle positive, we add 2\pi to it.

In the exercise, after converting \(-\frac{7\pi}{6}\) to the coterminal angle \(\frac{5\pi}{6}\), we can proceed with finding its reference angle.

Quadrants

The coordinate plane is divided into four quadrants, which helps in determining the direction and position of angles.

• Quadrant I: 0 to \frac{\pi}{2} (0° to 90°)
• Quadrant II: \frac{\pi}{2} to \pi (90° to 180°)
• Quadrant III: \pi to \frac{3\pi}{2} (180° to 270°)
• Quadrant IV: \frac{3\pi}{2} to 2\pi (270° to 360°)

The angle \frac{5\pi}{6} lies in the second quadrant (between \frac{\pi}{2} and \pi).

In the second quadrant, the reference angle is found by subtracting the given angle from \pi. So, for \frac{5\pi}{6}, we find the reference angle as follows:

\(\pi - \frac{5\pi}{6} = \frac{6\pi}{6} - \frac{5\pi}{6} = \frac{\pi}{6}\)

So, the reference angle for \(-\frac{7\pi}{6}\) is \(\frac{\pi}{6}\).