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Chapter 5: Problem 3

Convert each angle to radians. \(-45^{\circ}\)

Short Answer

Expert verified
The given angle \(-45^{\circ}\) is equal to \(-\frac{1}{4}\pi\) radians.

Step by step solution

01

Write the given angle

The given angle is \(-45^{\circ}\).
02

Apply the conversion factor

To convert the angle from degrees to radians, we need to multiply it by the conversion factor \(\frac{\pi \text{ radians}}{180^{\circ}}\). \(-45^{\circ}\cdot\frac{\pi \text{ radians}}{180^{\circ}}\)
03

Simplify the expression

To simplify, we can divide both numerators and denominators by their greatest common divisor (in this case, 45): \[\frac{-45}{180}\cdot\frac{\pi}{1} =\frac{-1}{4}\cdot\pi\]
04

Write the final answer

The given angle of \(-45^{\circ}\) is equal to \(-\frac{1}{4}\pi\) radians.

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Key Concepts

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

Understanding Radians
Radians are a fundamental unit for measuring angles, primarily used in mathematics and science. Unlike degrees, which break a circle into 360 parts, radians provide a measure based on the radius of the circle itself. Specifically, a radian is the angle formed when the arc length is equal to the radius. This relationship makes them extremely useful in various mathematical calculations and mathematical modeling.
  • 1 radian is approximately 57.3 degrees.
  • A full circle is 2蟺 radians, equivalent to 360 degrees.
  • This means half a circle is 蟺 radians, or 180 degrees.
The radian measure is favored in higher mathematics because it simplifies many formulas. For example, in trigonometry, calculus, and related fields, using radians removes the need for conversion factors when calculating derivatives and integrals.
Measurement in Degrees
Degrees are a more familiar way of measuring angles for most people, often used in everyday scenarios like navigation and simple geometry. A full circle, by definition, is exactly 360 degrees, a convention believed to date back to the Babylonians, who used a base-60 numeral system.
  • Degrees divide a circle into 360 equal parts.
  • They offer an intuitive understanding of angles, such as right angles being 90 degrees.
  • Angles in degrees are easy to understand and visualize.
Though practical for many common applications, degrees do not naturally fit situations requiring precise mathematical computation, where radians are more suitable. However, converting an angle from degrees to radians is straightforward with the right conversion factor.
The Role of the Conversion Factor
Converting between radians and degrees requires a consistent method to ensure accuracy, and that is done using a conversion factor. This vital factor, \(\frac{\pi \text{ radians}}{180^\circ}\), relies on the established relationships within a circle regarding radians and degrees.

How to Use the Conversion Factor

To convert from degrees to radians:
  • Multiply the angle in degrees by \(\frac{\pi}{180}\).
  • This changes the unit from degrees to radians.
To convert from radians to degrees, you simply use the reciprocal of that conversion factor:
  • Multiply the angle in radians by \(\frac{180}{\pi}\).
  • This changes the unit from radians to degrees.
For example, to convert \(-45^{\circ}\) to radians, you multiply \(-45\) by \(\frac{\pi}{180}\), simplifying using common factors. The result is \-\frac{1}{4}\pi\ radians, a clear indication of the conversion factor's usefulness in diverse mathematical contexts.

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