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Chapter 2: Problem 21

Name the reference angle \(\theta_{r}\) for the angle \(\theta\) given. $$\theta=-168.4^{\circ}$$

Short Answer

Expert verified
The reference angle is \(11.6^{\circ}\).

Step by step solution

01

Understand the concept of reference angle

The reference angle \(\theta_r\) is the smallest angle between the given angle \(\theta\) and the x-axis. It is always positive and is less than or equal to \(90^{\circ}\). Reference angles are always measured in the first quadrant.
02

Adjust for a positive angle

Since the given angle \(\theta = -168.4^{\circ}\) is negative, we need to find its positive equivalent within the standard \(0^{\circ} - 360^{\circ}\) range. This is done by adding \(360^{\circ}\) to the negative angle: \(-168.4^{\circ} + 360^{\circ} = 191.6^{\circ}\).
03

Determine the correct quadrant

The positive angle \(191.6^{\circ}\) lies in the third quadrant because it falls between \(180^{\circ}\) and \(270^{\circ}\). Angles in the third quadrant have their reference angles measured from \(180^{\circ}\).
04

Calculate the reference angle

For angles in the third quadrant, the reference angle \(\theta_r\) is \(\theta - 180^{\circ}\). Hence, the reference angle is \(191.6^{\circ} - 180^{\circ} = 11.6^{\circ}\).

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Key Concepts

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

Positive Equivalent Angle: Converting Negative Angles
When dealing with angles, it's common to encounter negative angles. They simply indicate a clockwise rotation, as opposed to a counterclockwise one for positive angles. To work with these in a standard context, it's often useful to find what's called the 'positive equivalent angle'. This essentially translates a negative angle into a positive measure that fits within the conventional circle of angles: 0掳 to 360掳.

Here's how you can find a positive equivalent angle:
  • Start with your given negative angle. For instance, in our problem, it's \(-168.4^{\circ}\).
  • Add \(360^{\circ}\) to this angle. This shifts the angle into the positive range while maintaining the same directional rotation on the circle.
  • Thus, \(-168.4^{\circ} + 360^{\circ} = 191.6^{\circ}\).
This positive equivalent angle, \(191.6^{\circ}\), can now be used in further calculations and is essential for tasks like quadrant identification and reference angle calculation.
Quadrant Identification: Locating Your Angle
Understanding which quadrant an angle lies in is crucial for determining the reference angle, as well as for other trigonometric calculations. The four quadrants of a circle are divided by angles at the axes:
  • First Quadrant: 0掳 to 90掳
  • Second Quadrant: 90掳 to 180掳
  • Third Quadrant: 180掳 to 270掳
  • Fourth Quadrant: 270掳 to 360掳
For our example, the positive equivalent angle was found to be \(191.6^{\circ}\). Since this value falls between \(180^{\circ}\) and \(270^{\circ}\), we identify that our angle is located in the third quadrant.

This is important because the reference angle is calculated differently in each quadrant, depending on its relation to the nearest axis. In the third quadrant, this involves subtracting \(180^{\circ}\) from the angle.
Angle Measurement: Calculating the Reference Angle
The reference angle is a fundamental concept in trigonometry. It's a measure of an angle's distance from the x-axis and is always between 0掳 and 90掳, ensuring it's positive and acute.

Here's how to find the reference angle for a given angle:
  • If the angle is in the First Quadrant, the reference angle is the angle itself.
  • If it's in the Second Quadrant, use \(180^{\circ} - \theta\).
  • In the Third Quadrant, subtract \(180^{\circ}\) from the angle. For \(191.6^{\circ}\), this calculation is \(191.6^{\circ} - 180^{\circ} = 11.6^{\circ}\).
  • Finally, in the Fourth Quadrant, use \(360^{\circ} - \theta\).
In our exercise, the angle \(191.6^{\circ}\) was in the third quadrant, leading us to a reference angle of \(11.6^{\circ}\). This measure is key for trigonometric calculations because it standardizes angles, allowing for easier analysis and application.

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Most popular questions from this chapter

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