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Chapter 8: Problem 16

Construct angles with the following radian measure. $$-\pi / 4,-3 \pi / 2,-3 \pi$$

Short Answer

Expert verified
-\( \pi / 4 \) places the terminal side in quadrant IV, -3\( \pi / 2 \) places it on the positive y-axis, and -3\( \pi \) places it on the positive x-axis.

Step by step solution

01

Understanding Negative Radian Measures

Negative radian measures indicate that the angle is measured clockwise from the positive x-axis.
02

Convert -\( \pi / 4 \)

-\( \pi / 4 \) radians means rotating \( \pi / 4 \) radians clockwise. Since \( \pi / 4 \) radians is equivalent to 45 degrees, -\( \pi / 4 \) is 45 degrees clockwise from the positive x-axis. This places the terminal side in the fourth quadrant.
03

Convert -3\( \pi / 2 \)

-3\( \pi / 2 \) radians is equivalent to three-quarters of a full rotation clockwise. Since \( 2\pi \) radians is a full rotation (360 degrees), three-quarters of this is 270 degrees. Hence, -3\( \pi / 2 \) radians will land you on the positive y-axis, specifically in the third quadrant.
04

Convert -3\( \pi \)

-3\( \pi \) radians means rotating 3\( \pi \) radians clockwise. Since each \( \pi \) radian is equivalent to 180 degrees, 3\( \pi \) radians is 540 degrees (or 1 and a half full rotations). This calculation lands you on the positive x-axis.

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Key Concepts

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

Radian Measures
Radian measures are a way to express angles based on the radius of a circle. Unlike degrees, which split a circle into 360 parts, radians relate directly to the arc length of the circle.

A full circle is 360 degrees, which is also equivalent to 2蟺 radians. This means:
  • 1 full circle = 2蟺 radians = 360 degrees
  • 1 radian 鈮� 57.3 degrees
  • 蟺 radians = 180 degrees
When dealing with negative radian measures, remember that these angles are measured clockwise from the positive x-axis. For instance, -蟺/4 means you rotate 蟺/4 radians in the clockwise direction, which places you at a 45-degree angle in the fourth quadrant.
Angle Conversion
Converting between radians and degrees is a crucial skill in trigonometry and related fields. To convert radians to degrees, you can use the relationship that 蟺 radians is equivalent to 180 degrees. This can be summarized through the following formula:
  • Degrees = Radians 脳 (180/蟺)
  • Radians = Degrees 脳 (蟺/180)
For example:

1. For -蟺/4 radians:
  • Degrees = (-蟺/4) 脳 (180/蟺) = -45 degrees


2. For -3蟺/2 radians:
  • Degrees = (-3蟺/2) 脳 (180/蟺) = -270 degrees


3. For -3蟺 radians:
  • Degrees = (-3蟺) 脳 (180/蟺) = -540 degrees
Converting angles helps you better understand their positions on the unit circle.
Circular Motion
Circular motion involves objects moving along a circular path. One key aspect is understanding the connection between angles (in radians or degrees) and the position on this path.

Consider an object moving around a circle with a radius r:
  • Positive angles mean counterclockwise rotation
  • Negative angles mean clockwise rotation
This is akin to tracing out paths on the unit circle, where:
  • -蟺/4 radians places the terminal side in the fourth quadrant
  • -3蟺/2 radians results in the terminal side on the positive y-axis
  • -3蟺 radians circles around one full rotation and then half, landing on the positive x-axis
Understanding circular motion and angles revolutionizes how we approach problems in trigonometry, physics, and engineering.

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