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Chapter 1: Problem 33

Convert each angle measure from degrees to radians. $$ 15^{\circ} $$

Short Answer

Expert verified
The angle of \(15^\circ\) in radians is \(\frac{\pi}{12}\).

Step by step solution

01

Conversion method

To convert from degrees to radians, we use the fact that \(180^\circ\) is equal to \(\pi\) radians. We can therefore create a conversion fraction of \(\frac{\pi}{180}\) radians per degree.
02

Apply Conversion to the Given Angle

To convert \(15^\circ\) into radians, we multiply it by the conversion fraction from step 1. So, the conversion will look like this: \(15^\circ \times \frac{\pi}{180}\).
03

Simplification

After performing the multiplication in Step 2, we simplify the resulting expression by dividing both numerator and denominator by 15, leaving us with the final answer: \(\frac{\pi}{12}\) radians.

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Key Concepts

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

Degrees to Radians
Understanding the conversion from degrees to radians is important when dealing with angles in mathematics and real-world problems. **Degrees** and **radians** are two units for measuring angles. While degrees are often used in everyday contexts, radians are more prevalent in mathematical calculations and in sciences related to physics and engineering.

Here is the key concept to remember:
  • There are 360 degrees in a full circle.
  • A circle is also 2蟺 radians.
  • Thus, 180 degrees is equivalent to 蟺 radians.
This relationship establishes the base for converting any angle measured in degrees to its equivalent measure in radians.
Pi Radians
The term **pi radians** might sound abstract, but it is merely a connection between radians and degrees using the mathematical constant 蟺. Pi (蟺) represents about 3.14159 and is crucial in geometry and trigonometry.

When you hear "pi radians," think of this as a definitive measure:
  • 1 radian is the angle at the center of a circle made by an arc equal in length to the radius.
  • Thus, 蟺 radians is equivalent to 180 degrees because the arc is half the circle.
Pi provides a natural way of describing angles, as it directly connects to the properties of a circle. This way, radians become a bridge between linear arc measures and full circular angles.
Conversion Method
Converting degrees to radians requires a simple multiplication, thanks to the conversion method based on the relationship between degrees and radians.
  • First, remember the conversion factor: \[ \frac{\pi}{180} \text{ radians per degree} \]
  • To convert, multiply the degree measure by this conversion factor.
For example, to convert 15 degrees to radians, you multiply 15 by \(\frac{\pi}{180}\). This gives \[15 \times \frac{\pi}{180} = \frac{15\pi}{180}\]By simplifying \(\frac{15\pi}{180}\), dividing the numerator and the denominator by 15, you are left with \[\frac{\pi}{12} \text{ radians}.\]The method is straightforward, giving you a precise and quick way to switch between degrees and radians anytime you need.

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Most popular questions from this chapter

Convert each angle measure to DMS notation. $$ 220.43^{\circ} $$

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