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Chapter 1: Problem 21

Convert each radian measure to degree measure. $$ -6 \pi $$

Short Answer

Expert verified
-1080 degrees.

Step by step solution

01

Understand the Conversion Factor

To convert radians to degrees, use the conversion factor that states: 1 radian = 180/蟺 degrees.
02

Apply the Conversion Factor

Given -6 蟺 radians, multiply by 180/蟺 to convert to degrees. \[ -6 \frac{\text{蟺} \times 180}{\text{蟺}} \] Note how the 蟺 in the numerator and denominator cancel each other out.
03

Simplify the Expression

Now simplify the multiplication: \[ -6 \times 180 = -1080 \] So, -6 蟺 radians is equal to -1080 degrees.

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Key Concepts

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

trigonometry
Trigonometry deals with the relationships between the angles and lengths of triangles. It is a foundational mathematical discipline used in various fields such as engineering, physics, and architecture. The primary functions in trigonometry are sine, cosine, and tangent, which help in solving right-angled triangles. However, trigonometry extends beyond triangles to describe circular and oscillatory motion using angles measured in radians and degrees.
radian measure
A radian is a unit of angular measure used in many areas of mathematics. One radian is the angle formed when the arc length of a circle is equal to the radius of the circle. Since a full circle is comprised of an arc length equal to the circumference of the circle, which is \(2\text{蟺}\) times the radius, a full circle is \(2\text{蟺}\) radians. In simpler terms:

  • 1 full circle = \(2\text{蟺}\) radians
  • 陆 circle = \(\text{蟺}\) radians
  • 录 circle = \(\frac{\text{蟺}}{2}\) radians.
degree measure
The degree is another unit of measurement for angles, more familiar in everyday life. A full circle is divided into 360 equal parts, each of which is a degree. That means:
  • 1 full circle = 360 degrees
  • 陆 circle = 180 degrees
  • 录 circle = 90 degrees
Switching between radians and degrees can make certain calculations more intuitive depending on the context.
conversion factor
Converting between radians and degrees involves using a conversion factor. The key relationship you need to remember is:
  • 1 radian = \(\frac{180}{\text{蟺}}\) degrees.
  • Conversely, 1 degree = \(\frac{\text{蟺}}{180}\) radians.
For example, converting -6蟺 radians to degrees, we multiply by the conversion factor \(\frac{180}{\text{蟺}}\):
-6蟺 radians * \(\frac{180}{\text{蟺}}\) = -1080 degrees.
The 蟺 symbols cancel out, simplifying to -6 * 180 = -1080 degrees. This conversion factor is crucial for switching between these two units easily.

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