Problem 29 - Precalculus

Question

Convert from radians to degrees. $$\frac{\pi}{6}$$

Step-by-Step Solution

Answer

\( \frac{\pi}{6} \) radians is equivalent to 30 degrees.

Step 1: Understand the Conversion Factor

To convert from radians to degrees, we use the conversion factor where \( 180^{\circ} \) is equivalent to \( \pi \) radians. This means we multiply the radian measure by \( \frac{180}{\pi} \) to get the degree measure.

Step 2: Apply the Conversion Factor

Take the given radian measure, \( \frac{\pi}{6} \), and multiply it by the conversion factor \( \frac{180}{\pi} \): \[ \frac{\pi}{6} \times \frac{180}{\pi} \]

Step 3: Simplify the Expression

Cancel out \( \pi \) from the numerator and denominator: \[ \frac{1}{6} \times 180 \]

Step 4: Calculate the Result

Now, calculate \( \frac{180}{6} \), which simplifies to \( 30 \). So, the radian measure \( \frac{\pi}{6} \) converts to \( 30^{\circ} \).

Key Concepts

Conversion Factor

The conversion factor is an important tool that allows us to switch between two units of measurement—radians and degrees. Understanding this is key to converting between radians and degrees.

For circular measurements, radians and degrees provide two different ways to express angles. They relate through the comprehensive conversion factor that associates the complete circle:

• Radians: A circle is made up of \( 2\pi \) radians.
• Degrees: The same circle is equivalent to \( 360^{\circ} \).

This equivalence means that \( 360^{\circ} = 2\pi \) or in simpler terms, \( 180^{\circ} = \pi \) radians. To convert any given radian measure to degrees, you multiply the radian measure by \[\frac{180}{\pi}. \]This fraction is the conversion factor. It works by proportionally scaling the radian value to a degree measure. Simply put, it recalibrates the angle measurement from the radian scale into the degree scale by accounting for the difference in units.

Radians

Radians are a unit of angle measurement based on the radius of a circle. Radians provide a natural way of measuring angles based on the geometry of a circle.

Think of a circle's radius as a unit measure. When the length of this radius is wrapped around the circumference, it covers an angle of one radian. Thus, radians are inherently linked to the circle itself. Here's a simple breakdown:

• Definition: One radian is the angle made when the arc length equals the radius of the circle.
• Full Circle: As the circumference of a circle is \( 2\pi \times \) the radius, a full circle encompasses \( 2\pi \) radians.

The radian measure is widely used in mathematical analysis and in contexts where it is more natural to reference the radius of a circle instead of an arbitrary division of the circle (as with degrees).

In calculus and advanced mathematics, radians are especially favorable because derivative and integral calculations involving trigonometric functions become simpler when angles are measured in radians.

Degrees

Degrees are perhaps the more familiar unit for measuring angles, especially in everyday settings. This unit divides a circle into 360 parts, called degrees.

The concept of degrees comes from historical and practical use, often applied in navigation, geography, and various fields. Here are some key aspects about degrees:

• Division of Circle: The full circle is divided into 360 equal parts.
• Historical Roots: The choice of 360 likely stems from ancient Babylonians who used base-60 mathematics, having 360 as a divisor for the solar year's days.

Degrees make it easier for many people to visualize and discuss angle sizes in geometric problems or real-world applications. They also provide a quick and intuitive way to communicate angles in more casual or non-technical scenarios.

Even though radians might be the choice for mathematical calculations, degrees remain prevalent in everyday usage, making the conversion between the two critical for effective communication across different disciplines.