Problem 65 - A Graphical Approach to Precalculus with Limits

Question

Convert each degree measure to radians. Leave answers as rational multiples of \(\pi .\) $$60^{\circ}$$

Step-by-Step Solution

Answer

\( 60^{\circ} \) equals \( \frac{\pi}{3} \) radians.

Step 1: Understanding the Problem

To convert an angle from degrees to radians, we use the fact that 180 degrees is equivalent to \( \pi \) radians. Our task is to convert \( 60^{\circ} \) to radians and express the result as a rational multiple of \( \pi \).

Step 2: Setting Up the Conversion Ratio

Since \( 180^{\circ} = \pi \) radians, the conversion factor is \( \frac{\pi}{180} \). We'll multiply this conversion factor by the degree measure to convert to radians.

Step 3: Performing the Conversion

Multiply \( 60^{\circ} \) by the conversion factor: \(60^{\circ} \times \frac{\pi}{180^{\circ}} = \frac{60\pi}{180} \).

Step 4: Simplifying the Fraction

Now simplify \( \frac{60\pi}{180} \) by dividing both the numerator and the denominator by their greatest common divisor, which is 60:\[ \frac{60\pi}{180} = \frac{\pi}{3} \]

Step 5: Final Expression

The simplified expression \( \frac{\pi}{3} \) is the measure of \( 60^{\circ} \) in radians, expressed as a rational multiple of \( \pi \).

Key Concepts

Degree to Radian Conversion

When converting angles from degrees to radians, it's essential to understand the relationship between these two units. Both are ways to measure angles. The key to converting degrees to radians lies in knowing that 180 degrees equals \( \pi \) radians. This forms the basis of our conversion factor.
To transform an angle in degrees to radians, we can use the conversion factor \( \frac{\pi}{180} \). Simply multiply the degree measure by this fraction:

• For example, to convert \( 60^{\circ} \) to radians, we calculate \( 60 \times \frac{\pi}{180} \).
• Simplifying, we get \( \frac{60\pi}{180} \).

This method efficiently translates the angle from degrees to radians, allowing for further simplification, which we'll explore next.

Pi as Rational Multiple

Expressing a radian measure as a rational multiple of \( \pi \) is a common and useful practice in trigonometry. This method helps in maintaining exact values instead of decimal approximations, providing a precise and clear representation.
After converting \( 60^{\circ} \) into the fraction \( \frac{60\pi}{180} \), we aim to reduce it to its simplest form. The number \( \pi \) remains a constant in this fraction, making it a rational multiple of \( \pi \), which is helpful in various calculations and problem-solving scenarios.
By simplifying the fraction correctly, you achieve a cleaner expression where \( \pi \) is maintained as a vital part of the measurement. Let's go through how simplification is achieved in the next section.

Simplifying Fractions

Simplifying fractions involves reducing the fraction to its smallest possible numerator and denominator by dividing both by their greatest common divisor (GCD). This makes the fraction easier to work with and often more meaningful.
In our exercise, the fraction \( \frac{60\pi}{180} \) needs simplification. The GCD of 60 and 180 is 60. Hence, we divide both the numerator and the denominator by 60:

• Numerator: \( 60 \div 60 = 1 \)
• Denominator: \( 180 \div 60 = 3 \)

After simplification, we obtain \( \frac{\pi}{3} \). This simplifies the expression significantly, providing a neat answer that conveys the precise angle measurement in radians as \( 60^{\circ} = \frac{\pi}{3} \). Understanding how and when to simplify fractions helps in solving many mathematical problems efficiently.