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Chapter 6: Problem 4

Find the reference angle for the given angle. $$ \begin{array}{llll}{\text { (a) } 120^{\circ}} & {\text { (b) }-210^{\circ}} & {\text { (c) } 780^{\circ}}\end{array} $$

Short Answer

Expert verified
Reference angles: (a) 60掳 (b) 30掳 (c) 60掳.

Step by step solution

01

Understanding Reference Angles

The reference angle for any angle is the smallest angle that the given angle makes with the x-axis. It's always between 0 and 90 degrees and is positive.
02

Find Reference Angle for 120掳

Since 120掳 is in the second quadrant, we find the reference angle by subtracting it from 180掳. So, the reference angle is \( 180掳 - 120掳 = 60掳 \).
03

Find Reference Angle for -210掳

First, find a coterminal angle by adding 360掳: \(-210掳 + 360掳 = 150掳\). Since 150掳 lies in the second quadrant, the reference angle is \(180掳 - 150掳 = 30掳\).
04

Find Reference Angle for 780掳

First, reduce 780掳 by subtracting 360掳 twice: \(780掳 - 360掳 - 360掳 = 60掳\). Since 60掳 is already a reference angle, the reference angle for 780掳 is simply 60掳.

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Key Concepts

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

Understanding Angles in Quadrants
When you're learning about angles, it's important to know how they fit into the four different quadrants of a coordinate plane. Think of the quadrants as four sections of a graph divided by the x and y axes. Here's how they are laid out:
  • Quadrant I: Both x and y coordinates are positive, where angles range from 0掳 to 90掳.
  • Quadrant II: The x coordinate is negative, and the y coordinate is positive, with angles ranging from 90掳 to 180掳.
  • Quadrant III: Both x and y coordinates are negative, for angles from 180掳 to 270掳.
  • Quadrant IV: The x coordinate is positive, and the y coordinate is negative, ranging from 270掳 to 360掳.
Remember, these quadrants help to determine the properties of an angle, including its reference angle.
Exploring Coterminal Angles
Coterminal angles are fascinating because they share the same terminal side or end point on the coordinate plane. You can find these angles by adding or subtracting full rotations of 360掳 from your given angle.

For example, the angle \( -210^{\circ} \) in our exercise needs to be adjusted since it's negative. By adding \( 360^{\circ} \), you reach \( 150^{\circ} \), a positive coterminal angle.

Another example is \( 780^{\circ} \), which is greater than 360掳. Subtracting \( 360^{\circ} \) twice results in \( 60^{\circ} \), another coterminal angle. This coterminal nature makes understanding and using angles far easier.
Mastering Angle Reduction
Angle reduction is an essential technique for simplifying the process of finding reference angles. When dealing with angles larger than 360掳 or negative angles, angle reduction comes into play.
  • For positive angles much larger than 360掳, like \( 780^{\circ} \), you repeatedly subtract 360掳 until you find an equivalent angle within the 0掳 to 360掳 range.
  • For negative angles, you add 360掳 until the angle becomes positive and falls within the standard range.
Through angle reduction, you simplify the problem and can easily proceed to find the reference angle, helping clarify the position and size of the angle on the coordinate plane.

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Most popular questions from this chapter

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