Problem 14 - Precalculus

Question

Convert from degrees to radians. Leave the answers in terms of \(\pi\). $$60^{\circ}$$

Step-by-Step Solution

Answer

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

Step 1: Understanding the Relationship

To convert degrees to radians, we use the fact that 180 degrees is equal to \( \pi \) radians. This means \( 1^{\circ} = \frac{\pi}{180} \) radians.

Step 2: Applying the Conversion Factor

Multiply 60 degrees by the conversion factor to find the equivalent in radians: \( 60^{\circ} \times \frac{\pi}{180} = \frac{60\pi}{180} \) radians.

Step 3: Simplifying the Expression

Simplify \( \frac{60\pi}{180} \) by dividing both the numerator and the denominator by their greatest common divisor, which is 60: \( \frac{60\pi}{180} = \frac{\pi}{3} \). Thus, 60 degrees is equal to \( \frac{\pi}{3} \) radians.

Key Concepts

Degrees to Radians

When converting angles from degrees to radians, we need to understand the relationship between these two units of measurement. Degrees and radians are both ways to measure angles.
A full circle comprises 360 degrees, while its equivalent in radians is represented by a value of \(2\pi\). Therefore, we use the equivalence of 180 degrees being equal to \(\pi\) radians.
To find how many radians correspond to one degree, use the formula:

• For one degree: \(1^{\circ} = \frac{\pi}{180}\) radians.

This basic relationship becomes the foundation for any conversion from degrees to radians, such as in the case where you convert 60 degrees as shown in the original exercise.
By multiplying the degrees by \(\frac{\pi}{180}\), we effectively express the angle in terms of \(\pi\) radians, allowing us to perform mathematical operations easily in calculus and trigonometry.

Angle Measurement

Understanding angle measurement is crucial for working in fields involving geometry and trigonometry. Degrees and radians are simply different ways to measure how 'open' an angle is.
Degrees are more intuitive for many people because they are commonly taught from a young age. A "right angle" is 90 degrees, and a "straight angle" is 180 degrees.
Radian measure, however, is based on the radius of a circle. It gives us a way to express angles as the length of the arc that an angle subtends on the unit circle.
For instance:

• A full circle is \(2\pi\) radians, or 360 degrees.
• A semi-circle is \(\pi\) radians, or 180 degrees.
• A 90-degree angle, which is a quarter circle, equals \(\frac{\pi}{2}\) radians.

Switching from degrees to radians helps simplify mathematical calculations, especially those involving periodic functions like sine and cosine.

Conversion Factor

The conversion factor plays a vital role in translating between degrees and radians. It's essentially a multiplier used to convert one unit of measure into another, ensuring the two values are equivalent.
The standard conversion factor from degrees to radians is \(\frac{\pi}{180}\). This factor stems from the equivalence that 180 degrees equals \(\pi\) radians.
When converting, you simply multiply the number of degrees by this conversion factor to get the angle in radians:

• Angle in degrees \(\times\) \(\frac{\pi}{180}\) = Angle in radians

For example, converting 60 degrees into radians involves the calculation:

• \(60^{\circ} \times \frac{\pi}{180} = \frac{60\pi}{180}\)
• Simplify \(\frac{60\pi}{180}\) to get \(\frac{\pi}{3}\) radians.

By using this factor, students can seamlessly convert any angle measurement from degrees to radians, aiding in deeper mathematical understanding.