Problem 36 - Precalculus

Question

Convert from radians to degrees. $$\frac{7 \pi}{3}$$

Step-by-Step Solution

Answer

\( \frac{7\pi}{3} \) radians is equal to 420 degrees.

Step 1: Understand the Conversion Formula

To convert radians to degrees, we use the conversion formula: \( \text{degrees} = \text{radians} \times \frac{180}{\pi} \) because \( \pi \) radians is equivalent to 180 degrees.

Step 2: Apply the Formula

Substitute the given radian measure \( \frac{7\pi}{3} \) into the conversion formula. This gives us: \( \text{degrees} = \frac{7\pi}{3} \times \frac{180}{\pi} \).

Step 3: Simplify the Expression

To simplify, first cancel out the \( \pi \) in the numerator and the denominator: \( \text{degrees} = \frac{7}{3} \times 180 \). Then, calculate \( \frac{7}{3} \times 180 \).

Step 4: Perform the Division and Multiplication

Calculate \( \frac{180}{3} \), which equals 60. Then multiply 60 by 7 to get the final degree measure: \( 60 \times 7 = 420 \).

Step 5: Conclusion

The radian measure \( \frac{7\pi}{3} \) converts to 420 degrees using our conversion process.

Key Concepts

Angle Measurement

Angles are a fundamental component of geometry, helping us describe rotation and orientation. When discussing angle measurements, two main units are commonly used: degrees and radians.

• Degrees are perhaps the most familiar unit. A full circle is divided into 360 degrees. Each degree can be further divided into 60 minutes, and each minute into 60 seconds.
• Radians are a bit different. They naturally emerge from the properties of the circle. A radian measures the angle described when the radius is mapped along the edge of the circle, making it an inherently circle-centered unit.

The conversion tasks arise because different fields use different units. Engineering and navigation might rely more on degrees, while mathematics and physics often use radians for their natural coupling to properties of circles and periodic functions.

Radian Conversion Formula

Converting between radians and degrees is straightforward thanks to a key conversion formula:\[\text{degrees} = \text{radians} \times \frac{180}{\pi}\]This formula stems from the relationship that a full circle is characterized by 360 degrees or \(2\pi\) radians. Therefore, \(\pi\) radians corresponds exactly to 180 degrees.

When you need to convert an angle from radians to degrees, simply multiply the radian measure by \(\frac{180}{\pi}\). This takes advantage of the fact that the ratio \(\frac{180}{\pi}\) represents the degree measure per one radian. So for example, the radian angle \( \frac{7\pi}{3} \) converts to degrees via:\[\frac{7\pi}{3} \times \frac{180}{\pi}\]Cancelling out \(\pi\) leaves us with the fraction \(\frac{7}{3}\) multiplied by 180, resulting in 420 degrees.

Unit Circle

The unit circle is a powerful mathematical tool that aids in understanding angle measures and their conversions. This circle has a radius of one unit and is centered at the origin of a coordinate plane.

The beauty of the unit circle lies in its simplicity and universality. Here's how it relates to our angle conversions:

• Every angle measured in radians on the unit circle can be represented by the radian length on the circle's circumference.
• A complete trip around the circle corresponds to an angle of \(2\pi\) radians, which is equivalent to 360 degrees.
• The unit circle allows each point, denoted by (cos(θ), sin(θ)), to directly correspond to the angle θ in both radians and degrees.

By visualizing or mapping radian measures onto the unit circle, understanding how those radian measures translate to degrees becomes more intuitive. This not only simplifies the conversion process but also deepens comprehension of trigonometric functions and their relationships with angles.