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Chapter 5: Problem 18

Determine whether the angles in each given pair are coterminal. $$744^{\circ},-336^{\circ}$$

Short Answer

Expert verified
Yes, the angles 744掳 and -336掳 are coterminal because they both reduce to 24掳.

Step by step solution

01

- Understand Coterminal Angles

Coterminal angles are angles that share the same terminal side when drawn in standard position. This means they differ by an integer multiple of 360 degrees.
02

- Find the Standard Angle for 744掳

To find a coterminal angle within the range of 0掳 to 360掳, subtract 360掳 repeatedly from 744掳 until the result is within this range: \[744^\text{掳} - 360^\text{掳} = 384^\text{掳}\]\[384^\text{掳} - 360^\text{掳} = 24^\text{掳}\]So, the standard angle for 744掳 is 24掳.
03

- Find the Standard Angle for -336掳

Similarly, to find a coterminal angle for -336掳, add 360掳 repeatedly until the result is within the range of 0掳 to 360掳:\[-336^\text{掳} + 360^\text{掳} = 24^\text{掳}\]So, the standard angle for -336掳 is also 24掳.
04

- Compare the Standard Angles

Since both 744掳 and -336掳 reduce to the same standard angle of 24掳, they are coterminal.

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Key Concepts

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

angles in standard position
When we talk about angles in standard position, we mean that the angle's vertex is at the origin of a coordinate system, and its initial side lies along the positive x-axis. To visualize this clearly, imagine a circle with a radius extending from the origin to any point along the circumference. The starting line of the angle will always be on the x-axis pointing right. From there, we measure the angle counterclockwise for positive angles, and clockwise for negative angles. This uniform starting point helps us easily compare and understand the properties of different angles.
coterminal angles
Coterminal angles are simply angles that share the same terminal side when drawn in standard position. This happens because they differ by a multiple of 360 degrees, which is the full rotation around a circle. To find coterminal angles, you can add or subtract 360 degrees any number of times. To see if given angles like \(744^\text{掳}\) and \(-336^\text{掳}\) are coterminal, you first convert them into an angle between 0 and 360 degrees, also known as their standard angle. If two angles reduce to the same standard angle, they are coterminal.
angle reduction
Angle reduction helps simplify the understanding of large or small angles by bringing them within the standard range of 0掳 to 360掳. It ensures easier comparison and helps recognize if they are coterminal. For instance, to reduce \(744^\text{掳}\), you subtract 360掳 repeatedly:
  • First, \(744^\text{掳} - 360^\text{掳} = 384^\text{掳}\).
  • Then, \(384^\text{掳} - 360^\text{掳} = 24^\text{掳}\).
This means \(744^\text{掳}\) reduces to \(24^\text{掳}\). Similarly, for \(-336^\text{掳}\), add 360掳:
  • \(-336^\text{掳} + 360^\text{掳} = 24^\text{掳}\).
So, \(-336^\text{掳}\) also reduces to \(24^\text{掳}\). Therefore, \(744^\text{掳}\) and \(-336^\text{掳}\) are coterminal since they share the same standard angle.

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