Problem 14 - Algebra and Trigonometry

Question

\(13-24\) . Find the degree measure of the angle with the given radian measure. $$ \frac{11 \pi}{3} $$

Step-by-Step Solution

Answer

The angle measures 660 degrees.

Step 1: Understand the Problem

We are given an angle in radians, \( \frac{11\pi}{3} \), and need to convert it to degrees. Recall that \( 180^\circ = \pi \) radians.

Step 2: Use the Radian to Degree Conversion Formula

The formula to convert radians to degrees is \( \text{Degrees} = \text{Radians} \times \frac{180}{\pi} \). Substituting \( \frac{11\pi}{3} \) for Radians gives us: \( \frac{11\pi}{3} \times \frac{180}{\pi} \).

Step 3: Simplify the Expression

Start by cancelling \( \pi \) from the numerator and the denominator: \( \frac{11}{3} \times 180 \).

Step 4: Calculate the Degrees

Now, compute the multiplication: \( \frac{11}{3} \times 180 = 11 \times 60 = 660 \).

Step 5: Conclusion

The angle with radian measure \( \frac{11\pi}{3} \) converts to \( 660^\circ \).

Key Concepts

Radian Measure

Radian measure is a unit of angular measure used in mathematics. It is based on the radius of a circle. When you think of radian measure, you can visualize it as the angle created by taking the radius of a circle and wrapping it along the circle's circumference. This method of measuring angles is quite natural for calculations in trigonometry, calculus, and many fields of physics.

• One complete revolution around a circle corresponds to an angle of 2\(\pi\) radians.
• A straight line (or half a circle) corresponds to an angle of \(\pi\) radians.
• Smaller angles can be expressed as fractions or multiples of \(\pi\).

Radian measure is dimensionless, meaning it doesn't rely on external units like degrees. This often simplifies mathematical calculations. In our exercise, the angle \( \frac{11\pi}{3} \) is given in radians, which can be broken down into more intuitive rotations around the circle.

Degree Measure

Degree measure is another way to express angles, commonly used in navigation, engineering, and everyday settings. Degrees break a circle into 360 equal parts. This format can be easier to comprehend in non-mathematical contexts.

• One complete circle is 360 degrees.
• Half a circle, which is a straight line, is 180 degrees.
• Quarter of the circle, a right angle, is 90 degrees.

Degrees are often more intuitive when dealing with everyday problems or when visualizing angles in a typical setting, such as the corners of a square or the direction of a compass. Understanding the degree measure allows you to easily relate mathematical concepts to real-world scenarios.

Angle Conversion Formula

The angle conversion formula is a bridge between radian and degree measures. It allows one to translate angles from one unit to the other efficiently. The formula is expressed as:\[\text{Degrees} = \text{Radians} \times \frac{180}{\pi}\]Here's how it works:

• To convert radians to degrees, multiply the radian measure by \(\frac{180}{\pi}\).
• This factor comes from the fact that \(\pi\) radians equal 180 degrees.

In the exercise, converting \( \frac{11\pi}{3} \) radians into degrees involved substituting into the formula:\[\frac{11\pi}{3} \times \frac{180}{\pi} = 660\degree\]We first cancel \(\pi\) from both the numerator and the denominator, then multiply \(\frac{11}{3}\) by 180 to get the final degree measurement. This simple and systematic approach allows you to switch between angle measures effortlessly.