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Chapter 1: Problem 17

Find the degree measures of two positive and two negative angles that are coterminal with each given angle. $$ -45^{\circ} $$

Short Answer

Expert verified
315掳, 675掳, -405掳, -765掳.

Step by step solution

01

Understanding Coterminal Angles

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

Find Two Positive Coterminal Angles

To find the positive coterminal angles, add 360掳 to the given angle. First: -45掳 + 360掳 = 315掳Second: Add another 360掳: 315掳 + 360掳 = 675掳
03

Find Two Negative Coterminal Angles

To find the negative coterminal angles, subtract 360掳 from the given angle. First: -45掳 - 360掳 = -405掳Second: Subtract another 360掳: -405掳 - 360掳 = -765掳

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Key Concepts

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

positive angles
Positive angles are measured in a counterclockwise direction from the initial side of an angle. By adding 360掳 to a given angle, we shift it one full rotation further while keeping the direction counterclockwise.

For example, to find positive coterminal angles for -45掳, start by:
  • Adding 360掳: ewline-45掳 + 360掳 = 315掳
  • Adding another 360掳: ewline 315掳 + 360掳 = 675掳

Therefore, 315掳 and 675掳 are positive angles coterminal with -45掳.
negative angles
Negative angles are measured in the clockwise direction from the initial side of an angle. By subtracting 360掳 from a given angle, we shift it one full rotation back while maintaining the clockwise direction.

For finding negative coterminal angles for -45掳, we perform the following steps:
  • Subtracting 360掳: ewline -45掳 - 360掳 = -405掳
  • Subtracting another 360掳: ewline -405掳 - 360掳 = -765掳

Thus, -405掳 and -765掳 are negative angles coterminal with -45掳.
angle measure
Angle measure determines the rotation amount from the initial side to the terminal side. Angles are usually measured in degrees (掳). One complete rotation around a circle equals 360掳.

An angle that measures -45掳 indicates that it is rotated 45掳 clockwise. Adding or subtracting multiples of 360掳, the measure enables us to find different angles that share the same initial and terminal sides.

Here, -45掳 is the original angle. By adding and subtracting 360掳, we find coterminal angles like 315掳, 675掳, -405掳, and -765掳.
full rotation
A full rotation around a circle is equal to 360掳. This concept helps us understand how coterminal angles work. When you add or subtract 360掳 to any angle, you can find an angle that starts and ends at the same position on the circle.

Let's say we start with -45掳:
  • One full rotation added brings us to: -45掳 + 360掳 = 315掳
  • Another full rotation added brings us to: 315掳 + 360掳 = 675掳
  • One full rotation subtracted shows: -45掳 - 360掳 = -405掳
  • Another full rotation subtracted shows: -405掳 - 360掳 = -765掳

Hence, every addition or subtraction of 360掳 gives us coterminal angles.

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Most popular questions from this chapter

True or false? Do not use a calculator. $$ \cos (9 \pi / 5)=\cos (\pi / 5) $$

Find the radius of the circle in which the given central angle \(\alpha\) intercepts an arc of the given length s. Round to the nearest tenth. $$ \alpha=360^{\circ}, s=8 \mathrm{~m} $$

Perform the indicated operation. Express the result in terms of \(\pi\) $$ \frac{\pi}{2}+\frac{\pi}{4} $$

Find the exact area of the sector of the circle with the given radius and central angle. $$ r=4, \alpha=45^{\circ} $$

Find the radius of the circle in which the given central angle \(\alpha\) intercepts an arc of the given length s. Round to the nearest tenth. $$ \alpha=0.004, s=99 \mathrm{~km} $$
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