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Chapter 5: Problem 27

Determine two coterminal angles in degree measure (one positive and one negative) for each angle. (There are many correct answers). (a) \(300^{\circ}\) (b) \(-740^{\circ}\)

Short Answer

Expert verified
The positive coterminal angles for \(300^{\circ}\) and \(-740^{\circ}\) are \(660^{\circ}\) and \(-20^{\circ}\) respectively, and their negative coterminal angles are \(-60^{\circ}\) and \(-1100^{\circ}\) respectively.

Step by step solution

01

Find positive coterminal angle for \(300^{\circ}\)

Our goal here is to add 360 degrees to the given angle. When 360 degrees (full circle) is added to the angle \(300^{\circ}\), it would result in \(300^{\circ} + 360^{\circ} = 660^{\circ}\). Thus, \(660^{\circ}\) is a positive coterminal angle for \(300^{\circ}\).
02

Find negative coterminal angle for \(300^{\circ}\)

To find a negative coterminal angle, we subtract 360 degrees from the given angle. So, \(300^{\circ} - 360^{\circ} = -60^{\circ}\). Therefore, \(-60^{\circ}\) is a negative coterminal angle for \(300^{\circ}\).
03

Find positive coterminal angle for \(-740^{\circ}\)

In this case, we add 360 degrees to the given angle until we get a positive angle. By adding 360 twice, we get \(-740^{\circ} + 2*360^{\circ} = -20^{\circ}\). So, \(-20^{\circ}\) is a positive coterminal angle for \(-740^{\circ}\).
04

Find negative coterminal angle for \(-740^{\circ}\)

To find a negative coterminal angle, we subtract 360 degrees from the given angle. So, \(-740^{\circ} - 360^{\circ} = -1100^{\circ}\). Therefore, \(-1100^{\circ}\) is another negative coterminal angle for \(-740^{\circ}\).

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Key Concepts

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

Degree Measure
Understanding degree measure is fundamental in trigonometry. Degrees are a way to measure angles, and they tell us how far we've rotated from a starting point. Think of a circle. A full rotation around the circle equals 360 degrees. This means a right angle is 90 degrees, and a straight line is 180 degrees.

Angles are often described in terms of degrees, and this measurement helps us understand direction and position. For example, if you draw a line from the center of a circle to its edge, rotating that line one full circle around equals 360 degrees. Always remember: degrees open up the world of angles for understanding and calculation.
Positive Angle
A positive angle occurs when you rotate counterclockwise from a starting position. It's a directionality concept that's helpful in determining how angles interact on a plane.
  • When dealing with coterminal angles, adding full circles (360 degrees) will always give you a positive coterminal angle.
  • For instance, if you start with 300 degrees and want a positive coterminal angle, you add 360 degrees to get 660 degrees.

Positive angles are used to express an angle鈥檚 standard position. They are intuitive as long as you remember the counterclockwise direction!
Negative Angle
Negative angles are the opposite of positive angles. They're created by rotating clockwise, which is why they're important in trigonometry.

When you want a negative coterminal angle, you subtract full circles (360 degrees) from the given angle. For example:
  • Starting with 300 degrees, subtract 360 degrees to get -60 degrees, a negative coterminal angle.
  • For more negative angles, repeat the process (subtracting additional 360s).
Negative angles allow us to describe positions and rotations in both directions, adding flexibility to our understanding of trigonometric calculations.
Full Circle
The concept of a full circle is crucial when dealing with angles, especially coterminal angles. A full circle corresponds to 360 degrees. This means if you start at a certain point and rotate around to return to that same point, you've turned 360 degrees.
  • When finding coterminal angles, you can add or subtract multiples of 360 to reach different angles that share the same terminal side.
  • For angles like -740 degrees, adding 360-degree cycles helps reach a positive angle.

Remember, a full circle is a complete rotation, and understanding how it affects angles is key to mastering coterminal angles.

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Most popular questions from this chapter

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