Problem 122 - Calculus Volume 1

Question

For the following exercises, convert each angle in radians to degrees. $$\frac{5 \pi}{12} \mathrm{rad}$$

Step-by-Step Solution

Answer

\( \frac{5\pi}{12} \) radians is equal to 75 degrees.

Step 1: Understanding the Formula

To convert an angle from radians to degrees, use the conversion factor where 1 radian equals \( \frac{180}{\pi} \) degrees. This means that you multiply the radian measure by \( \frac{180}{\pi} \) to get the degree measure.

Step 2: Applying the Conversion

Given the angle \( \frac{5\pi}{12} \) radians, multiply it by \( \frac{180}{\pi} \) to convert to degrees: \[ \frac{5\pi}{12} \times \frac{180}{\pi} \]

Step 3: Simplifying the Equation

Now simplify the equation:\[ \frac{5 \pi \times 180}{12 \times \pi} \]Here, the \(\pi\) terms in the numerator and denominator cancel out:\[ \frac{5 \times 180}{12} \]

Step 4: Calculating the Result

Perform the arithmetic:\[ 5 \times 180 = 900 \]Then divide by 12:\[ \frac{900}{12} = 75 \] Thus, \( \frac{5\pi}{12} \) radians is equal to 75 degrees.

Key Concepts

Radians to Degrees

When working with angles, you often need to switch between radians and degrees. Radians and degrees are two ways to measure angles, just like inches and centimeters are units of length. One full circle is 360 degrees or \( 2\pi \) radians. Therefore, if you know an angle in radians, you can convert it to degrees.

The conversion between radians and degrees is crucial, especially in trigonometry and calculus. It helps compare and understand angles in different contexts. For instance, if you have an angle expressed in radians, to convert it to degrees, you need to use the conversion factor.

This simple conversion process makes calculations easier in various applications, including physics, engineering, and computer graphics. Understanding how to switch between these units will give you a better grasp of the subject.

Conversion Factor

The conversion factor is like a bridge that connects radians and degrees. This factor comes from the relationship where \( \pi \) radians equal 180 degrees. Hence, one radian is equivalent to \( \frac{180}{\pi} \) degrees. This means to convert radians to degrees, you multiply by \( \frac{180}{\pi} \).

Let's say you have an angle in radians, for example, \( \frac{5\pi}{12} \). By multiplying it with the conversion factor \( \frac{180}{\pi} \), you're essentially scaling the radians into degrees. The \( \pi \) terms cancel out during this multiplication process.

Knowing this conversion factor by heart is very useful in mathematical calculations and helps you navigate between radians and degrees with ease. Remember, learning the conversion factor assists in making geometric problems and even real-life calculations smoother.

Mathematical Simplification

Simplifying equations is part of solving mathematical problems efficiently. In our conversion problem, once you've applied the conversion factor, you end up with an equation that may look complex but is not.

For example, after applying the conversion factor to \( \frac{5\pi}{12} \):

• Multiply the radian measure by \( \frac{180}{\pi} \): \( \frac{5\pi \times 180}{12\times \pi} \)
• The \( \pi \) terms cancel out, simplifying this to \( \frac{5 \times 180}{12} \)

This simplification step gets rid of unnecessary parts and leaves you with just what's needed to find the answer.

Once simplified, the arithmetic is straightforward: \( 5 \times 180 = 900 \), and divide \( 900 \) by \( 12 \) to get \( 75 \). Understanding how to simplify effectively keeps calculations manageable and avoids unnecessary complexity, which is especially important in bigger, more complicated problems.