Problem 71 - A Graphical Approach to Precalculus with Limits

Question

Convert each radian measure to degrees. $$\frac{\pi}{3}$$

Step-by-Step Solution

Answer

\(\frac{\pi}{3}\) radians is 60 degrees.

Step 1: Understand the Conversion Formula

To convert radians to degrees, we use the conversion formula: degrees = radians × \( \frac{180}{\pi} \). This formula is derived from the fact that \( \pi \) radians is equal to 180 degrees.

Step 2: Substitute the Radian Value

We have \( \frac{\pi}{3} \) radians. Substitute this into the conversion formula: degrees = \( \frac{\pi}{3} \times \frac{180}{\pi} \).

Step 3: Simplify the Expression

We can simplify \( \frac{\pi}{3} \times \frac{180}{\pi} \) by canceling \( \pi \) in the numerator and the denominator. This leaves us with \( \frac{180}{3} \).

Step 4: Perform the Division

Now, divide 180 by 3 to find the degree measure. \( 180 \div 3 = 60 \). Therefore, \( \frac{\pi}{3} \) radians is equal to 60 degrees.

Key Concepts

Radian Measure

Radians are a way of measuring angles based on the radius of a circle. It's different from degrees, which are based on dividing a circle into 360 parts. A radian measures how far you've traveled around the circle in terms of radius lengths. To visualize this, imagine a circle where the distance around the edge is exactly the same as the length of the radius. When you wrap the radius around the circle's edge, the angle formed at the center by that arc is 1 radian.
Radians provide a natural way to describe angles in circles because they relate the arc length directly to the radius. By using radians, many mathematical formulas become simpler.
For example, the circumference of a circle is simply its radius times \( 2\pi \). Here, \( \pi \) is roughly 3.14159, which represents the angle in radians of a half-circle. Thus, a full circle is \( 2\pi \) radians.

Degree Measure

Degrees are the most common unit for measuring angles, often used in everyday contexts. A full circle is divided into 360 equal parts, each one degree. This ancient division likely comes from earlier observations of the stars, which are approximately 360 days in a year.
By using degrees, we have a more intuitive and easily understandable framework for measuring angles. You can think of 90 degrees as one-fourth of a full circle, corresponding to a right angle.
Different fractions of a circle have particular degree measures, like 180 degrees for a half-circle, which corresponds to a straight line. Degrees make it easier to discuss parts of a circle in everyday terms, breaking them down into simple fractions.

Conversion Formula

The conversion formula between radians and degrees is crucial for switching between these two measurement systems. It's given by:

• \( \text{Degrees} = \text{Radians} \times \frac{180}{\pi} \)
• \( \text{Radians} = \text{Degrees} \times \frac{\pi}{180} \)

This formula comes from the understanding that \( \pi \) radians equal 180 degrees. Thus, when converting, you multiply the radian measure by \( \frac{180}{\pi} \) to convert it to degrees.

For instance, if you have \( \frac{\pi}{3} \) radians, you would substitute into the formula to get:

• \( \text{Degrees} = \frac{\pi}{3} \times \frac{180}{\pi} \)
• Cancel \( \pi \), leaving \( \frac{180}{3} = 60 \) degrees

This cleanly demonstrates how radians can be converted to degrees using the formula, providing a straightforward method for quick conversions.