Problem 113 - Calculus Volume 1

Question

Convert each angle in degrees to radians. Write the answer as a multiple of \(\pi\). $$240^{\circ}$$

Step-by-Step Solution

Answer

The angle \( 240^{\circ} \) is equivalent to \( \frac{4\pi}{3} \) radians.

Step 1: Understand the Conversion Formula

To convert an angle from degrees to radians, use the formula \( \theta \, \text{radians} = \theta \, \text{degrees} \times \frac{\pi}{180} \). This formula derives from the proportion that 180 degrees is equal to \( \pi \) radians.

Step 2: Apply the Conversion Formula

Substitute \( 240^{\circ} \) into the conversion formula:\[ 240^{\circ} \times \frac{\pi}{180} \] This expression will convert 240 degrees to radians.

Step 3: Simplify the Expression

Calculate the multiplication: \[ 240 \times \frac{\pi}{180} = \frac{240\pi}{180} \]Next, simplify \( \frac{240}{180} \) by dividing both the numerator and the denominator by their greatest common divisor, which is 60.

Step 4: Simplify the Fraction

Divide the numerator and denominator by 60:\[ \frac{240}{180} = \frac{240 \div 60}{180 \div 60} = \frac{4}{3} \]Thus, the expression becomes \( \frac{4\pi}{3} \).

Step 5: Finalize the Answer

The radian measure of \( 240^{\circ} \) is \( \frac{4\pi}{3} \). Write this as your final answer, as it is already in its simplest form.

Key Concepts

Degrees to Radians

Angle measure is a fundamental concept in mathematics and is often expressed in degrees. However, in trigonometry, radians are more useful. This concept helps to link angle measures with arc lengths and other properties in circles. So, how do we convert degrees to radians? It's simple! The conversion formula is \( \theta \, \text{radians} = \theta \, \text{degrees} \times \frac{\pi}{180} \). This means every one degree equals \( \frac{\pi}{180} \) radians.
For instance, converting \( 240^{\circ} \) to radians involves straightforward substitution into the formula:

• \( 240^{\circ} \times \frac{\pi}{180} \)

Through some basic simplification, this converts the angle from the degree measure to radian measure, offering a more efficient angle representation in many mathematical contexts.

Trigonometric Conversion

Trigonometric functions such as sine, cosine, and tangent often require angles in radians for calculations. This is because radians align more naturally with the properties of trigonometric functions.
When you begin a trigonometric conversion, ensure your angles are in radians. One complete circle in radians is \( 2\pi \). This aligns seamlessly with the wave-like behavior of the trigonometric functions.
For our example, converting \( 240^{\circ} \) to \( \frac{4\pi}{3} \) radians brings your angle neatly within this framework:

• This means the angle can be directly plugged into trigonometric functions.
• Accurate trigonometric values can then be calculated.

Thus, using radians simplifies and streamlines trigonometric calculations.

Mathematical Fractions

Converting angles doesn't just involve changing units; it also often involves working with fractions. Fractions are mathematical expressions that represent the ratio of one quantity to another. In our example, after converting, we obtained \( \frac{240\pi}{180} \), which needed simplifying.
The process of simplifying involves finding the greatest common divisor (GCD) between two numbers and dividing both by it. Here, 60 is the GCD of 240 and 180:

• Divide the numerator \(240\) by 60.
• Divide the denominator \(180\) by 60 as well.

This results in \( \frac{4}{3} \), giving us a simplified fraction of \( \frac{4\pi}{3} \).
Not only does this make the expression cleaner, it also facilitates easier use in further mathematical operations, ensuring precision and simplicity.