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Chapter 1: Problem 52

Find the degree measure of the smallest positive angle that is coterminal with each angle. $$ 1235.6^{\circ} $$

Short Answer

Expert verified
155.6掳

Step by step solution

01

Understand Coterminal Angles

Coterminal angles are angles that share the same initial and terminal sides. To find a coterminal angle, add or subtract multiples of 360掳 to the given angle.
02

Subtract 360掳 Until Positive Angle Less Than 360掳

Given angle: 1235.6掳. Subtract 360掳 repeatedly until the angle is less than 360掳. 1235.6掳 - 360掳 = 875.6掳 875.6掳 - 360掳 = 515.6掳 515.6掳 - 360掳 = 155.6掳
03

Identify the Smallest Angle

After subtracting 360掳 three times, the smallest positive angle that is coterminal with 1235.6掳 is 155.6掳.

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Key Concepts

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

angle measure
Understanding angle measures is crucial in trigonometry. An angle is formed by two rays meeting at a common endpoint called the vertex. The size of an angle can be specified in degrees or radians.

Angles in trigonometry are usually measured in degrees, where a full circle is 360 degrees. So, any angle measurement will fall between 0掳 and 360掳 to represent one full circle.

To visualize, imagine a clock. If the minute hand completes one round, it moves through 360掳. This concept helps us understand how angles work and how coterminal angles are derived.
degree measure
The degree measure of an angle tells us how far one of its rays has 'rotated' around the vertex from its starting position.

When discussing coterminal angles, understanding degree measures becomes even more important. For example, in the exercise given, 1235.6 degrees seems much larger than a full circle (360 degrees).

Coterminal angles help us to reduce large angles into a basic form that is easy to understand and compare. We can find a smaller angle that shares the same terminal side by repeatedly subtracting 360掳 until the angle falls within the range of 0掳 to 360掳.
trigonometry problem-solving
Trigonometry involves solving problems related to angles and their measures. A common task is finding coterminal angles. This helps simplify calculations and understand relationships between different angles.

In our example, we found a small positive coterminal angle for 1235.6掳. The process involved subtracting 360掳 multiple times:

  • 1235.6掳 - 360掳 = 875.6掳
  • 875.6掳 - 360掳 = 515.6掳
  • 515.6掳 - 360掳 = 155.6掳

This technique is handy for breaking down complex problems in trigonometry and showing that large and cumbersome angles can be made straightforward by working through them step-by-step.

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Most popular questions from this chapter

Fifteen experts are voting to determine the best convertible of the year. The choices are a Porsche Carrera, a Chrysler Crossfire, and a Nissan Roadster. The experts will rank the three cars 1 st, 2nd, and 3 rd. There are three common ways to determine the winner: a. Plurality: The car with the most first-place votes (preferences) is the winner. b. Instant runoff: The car with the least number of preferences is eliminated. Then the ballots for which the eliminated car is first are revised so that the second-place car is moved to first. Finally, the car with the most preferences is the winner. c. The point system: Two points are given for each time a car is ranked first place on a ballot, one point for each time the car appears in second place on a ballot, and no points for third place. When the ballots were cast, the Porsche won when plurality was used, the Chrysler won when instant runoff was used, and the Nissan won when the point system was used. Determine 15 actual votes for which this result would occur.

True or false? Do not use a calculator. $$ \sin \left(91^{\circ}\right)=-\sin \left(89^{\circ}\right) $$

True or false? Do not use a calculator. $$ \sin \left(179^{\circ}\right)=-\sin \left(1^{\circ}\right) $$

A point on Earth's equator makes one revolution in 24 hours. Find the linear velocity in feet per second for such a point, using 3950 miles as the radius of Earth.

True or false? Do not use a calculator. $$ \cos \left(150^{\circ}\right)=-\cos \left(30^{\circ}\right) $$
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