Problem 32 - Precalculus

Question

For the following exercises, convert angles in radians to degrees. \(\frac{11 \pi}{6}\) radians

Step-by-Step Solution

Answer

\(\frac{11 \pi}{6}\) radians is 330 degrees.

Step 1: Understand the Conversion Formula

To convert an angle from radians to degrees, use the formula: \( ext{Degrees} = ext{Radians} \times \frac{180}{ ext{π}} \). This formula is derived from the fact that \( ext{π} \) radians is equivalent to 180 degrees.

Step 2: Substitute Given Radians

Substitute \( \frac{11 ext{π}}{6} \) for Radians in the conversion formula: \( ext{Degrees} = \frac{11 ext{π}}{6} \times \frac{180}{ ext{π}} \).

Step 3: Simplify the Expression

First, cancel out \( ext{π} \) from the numerator and denominator of the fraction. This leaves us with \( \frac{11}{6} \times 180 \). Next, calculate the multiplication.

Step 4: Calculate the Result

Multiply \( \frac{11}{6} \) by 180, which results in \( 11 \times 30 = 330 \). Therefore, \( \frac{11 ext{π}}{6} \) radians is equal to 330 degrees.

Key Concepts

Radians to Degrees

Understanding how to convert angles measured in radians to degrees is essential for many mathematical applications. Angles can be expressed in degrees or radians, but depending on the context, one might be more convenient than the other. Here's a simple illustration of the conversion process. Radians and degrees are both units for measuring angles. One full circle is represented as either 360 degrees or \(2\pi\) radians. This means that \(\pi\) radians is equal to 180 degrees. By understanding this relationship, we can convert radians to degrees easily. Knowing how these measurements relate simplifies solving geometry and trigonometry problems.For instance, if we have an angle measuring \(\frac{11 \pi}{6}\) radians, we can convert it to degrees to understand its size relative to a complete circle. This conversion comes in handy across various scientific and engineering fields.

Conversion Formula

The conversion from radians to degrees uses a straightforward formula, which helps bridge the gap between these two measurements. Here's a breakdown of how it works:The formula to convert an angle from radians to degrees is:

• \( \text{Degrees} = \text{Radians} \times \frac{180}{\pi} \)

This means you multiply the radian measure by \(\frac{180}{\pi}\) to find its equivalent in degrees. Let's apply this to our example, \( \frac{11\pi}{6} \) radians. By substituting this value into the formula, we have:

• \( \text{Degrees} = \frac{11\pi}{6} \times \frac{180}{\pi} \)

This setup prepares us for the next step, which is simplifying the expression to reach a final result.

Simplifying Expressions

After setting up the conversion formula, the next step involves simplifying the expression to find the angle in degrees. Here's how you simplify:First, notice that \(\pi\) appears both in the numerator and the denominator. This allows us to cancel out \(\pi\) from the expression:

• \( \frac{11\pi}{6} \times \frac{180}{\pi} = \frac{11}{6} \times 180 \)

Now we no longer have \(\pi\) in our expression, and what's left is a simple multiplication.To get the final degree measure, calculate \( \frac{11}{6} \times 180 \). This involves multiplying 11 by 30 (since \( \frac{180}{6} = 30 \)), which gives us:

• \( 11 \times 30 = 330 \)

Thus, \(\frac{11\pi}{6}\) radians equals 330 degrees. Simplifying expressions in this manner not only aids in conversion but also enhances the clarity and accuracy of mathematical communication.