Problem 16 - Precalculus Mathematics for Calculus

Question

Find the degree measure of the angle with the given radian measure. $$\frac{11 \pi}{3}$$

Step-by-Step Solution

Answer

The angle is 660 degrees.

Step 1: Convert Radian Measure to Degree Measure

To convert a radian measure to a degree measure, use the formula: \[ ext{Degrees} = ext{Radians} imes rac{180}{ ext{π}} \]Plug the given radian measure, \( \frac{11\pi}{3} \), into this formula: \[ ext{Degrees} = \frac{11\pi}{3} \times \frac{180}{\pi} \]

Step 2: Cancel Out π and Simplify

Cancel the \(\pi\) from both the numerator and the denominator: \[ ext{Degrees} = \frac{11}{3} \times 180 \]Now, perform the multiplication: \[ = \frac{11 imes 180}{3} \]

Step 3: Calculate the Final Degree Measure

First, simplify the expression by dividing 180 by 3:\[ 180 \div 3 = 60 \]Then multiply by 11:\[ 11 imes 60 = 660 \]Therefore, \( \frac{11\pi}{3} \) radians is equal to 660 degrees.

Key Concepts

Radian Measure

Radians are a fundamental way to measure angles, primarily used in trigonometry and calculus. One radian is defined as the angle created at the center of a circle by an arc whose length is equal to the radius of the circle. Unlike degrees, which divide a circle into 360 parts, radians are based on the properties of the circle itself. A full circle has an angle of 2\(\pi\) radians, which approximately translates to 6.28318 radians. This means a radian is just another way to express angle measurements, offering a more natural relation to circular motion.

Here's a quick breakdown of key points about radians:

• One full circle is 2\(\pi\) radians.
• Half a circle (or a straight angle) is \(\pi\) radians.
• A right angle is \(\frac{\pi}{2}\) radians.

Understanding radians helps significantly with solving problems involving periodic functions and motions, like waves or rotations.

Degree Measure

Degrees are the most common unit of measurement for angles, widely favored for their ease of understanding and use in navigation, construction, and geometry. A degree splits a whole circle into 360 equal pieces. The number 360 is convenient due to its many divisors, facilitating practical daily calculations.

Key highlights of degree measure include:

• A complete circle is 360 degrees.
• A half-circle, or a straight line, is 180 degrees.
• A right angle, commonly seen in squares and rectangles, is 90 degrees.

This system is intuitive for many because it relates closely to the way humans perceive and describe directions and angles in everyday life. For scientific and engineering work, though, radians might offer more practical advantages.

Angle Conversion

Converting between radians and degrees is an essential skill in mathematics, especially when solving trigonometric problems or working in various applied sciences. The conversion is based on the relationship that 180 degrees equals \(\pi\) radians. Thus, any angle in radians can be converted to degrees by multiplying by \(\frac{180}{\pi}\), and vice versa by using \(\frac{\pi}{180}\).

For hands-on conversions, consider these examples:

• To convert \(\frac{11\pi}{3}\) radians to degrees, multiply by \(\frac{180}{\pi}\), cancel \(\pi\), and compute: \(\frac{11}{3} \times 180 = 660\) degrees.
• To convert 90 degrees to radians, use \(90 \times \frac{\pi}{180} = \frac{\pi}{2}\) radians.

These conversions are crucial in fields where different units may interchange frequently, requiring precise and swift swapping between units.