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Chapter 7: Problem 54

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

Short Answer

Expert verified
The reference angle is \( \frac{\text{蟺}}{6} \).

Step by step solution

01

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 \(\frac{\text{蟺}}{2}\) radians.
02

Convert the Negative Angle to a Positive Angle

Add 2蟺 to the given angle \(-\frac{7\text{蟺}}{6}\) since an angle plus 2蟺 is coterminal with the original angle and will lie in the same position. \(-\frac{7\text{蟺}}{6} + 2\text{蟺} = \ -\frac{7\text{蟺}}{6} + \frac{12\text{蟺}}{6} = \frac{5\text{蟺}}{6} \)
03

Identify the Quadrant of the Angle

Since \(\frac{5\text{蟺}}{6}\) is in the second quadrant (because it is between \(\frac{\text{蟺}}{2}\) and \(\text{蟺}\)), the reference angle is \(\text{蟺}\) minus the angle.
04

Calculate the Reference Angle

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

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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


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 \(\frac{\pi}{6}\).

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Most popular questions from this chapter

Use the definition or identities to find the exact value of each of the remaining five trigonometric functions of the acute angle \(\theta\). $$\cos \theta=\frac{1}{3}$$

Use a calculator to find the approximate value of each expression. Round the answer to two decimal places. $$ \tan 1 $$

Name the quadrant in which the angle \(\theta\) lies. $$ \cos \theta>0, \quad \cot \theta<0 $$

Convert each angle to a decimal in degrees. Round your answer to two decimal places. \(61^{\circ} 42^{\prime} 21^{\prime \prime}\)

Convert each angle in degrees to radians. Express your answer in decimal form, rounded to two decimal places. \(350^{\circ}\)
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