Problem 40 - Algebra 2

Question

Rewrite each degree measure in radians and each radian measure in degrees. \(660^{\circ}\)

Step-by-Step Solution

Answer

\(660^{\circ}\) is \(\frac{11\pi}{3}\) in radians.

Step 1: Know the Conversion Formula

To convert degrees to radians, use the formula \( \text{radians} = \text{degrees} \times \frac{\pi}{180} \).

Step 2: Apply the Formula

Substitute \(660^{\circ}\) into the formula: \[ \text{radians} = 660 \times \frac{\pi}{180} \].

Step 3: Simplify the Expression

Calculate the multiplication: \[ 660 \times \frac{\pi}{180} = \frac{660\pi}{180} \].Simplify the fraction: \( \frac{660}{180} \) which reduces to \( \frac{11}{3} \).

Step 4: Final Result in Radians

The radian measure for \(660^{\circ}\) is \( \frac{11\pi}{3} \).

Key Concepts

Radian Measure

Understanding radian measure is crucial when working with angles. A radian is an angle measure based on the radius of a circle. Think of it like this: if you take the radius of a circle and wrap it around the circumference, the angle subtended at the center by this arc is one radian. This is a way of expressing angles that is more natural in many branches of mathematics, especially calculus. Unlike degrees, which divide a circle into 360 parts, radians are based on the arc length. In mathematical terms, a full circle has a radian measure of \(2\pi\), meaning that \(360^{\circ} = 2\pi\) radians. This makes one radian equivalent to \(\frac{180}{\pi}\) degrees.

Degree Measure

Degrees are the most commonly used unit for measuring angles. They divide a circle into 360 equal parts. So, one degree represents \(\frac{1}{360}\)th of a full circle. Using degrees is helpful in many practical applications where simpler arithmetic calculations are involved. When dealing with geometry, navigation, and various engineering disciplines, degree measure is preferred. Degrees are easy to visualize, which is why they are taught early in math education.

Conversion Formula

Converting between radians and degrees requires a simple formula. This formula ensures precision and consistency when changing units. Here’s the formula used for conversion:

• To convert from degrees to radians, multiply the degree value by \(\frac{\pi}{180}\). For example, converting \(660^{\circ}\) would be achieved with the calculation \(660 \times \frac{\pi}{180}\).
• To convert from radians to degrees, multiply the radian value by \(\frac{180}{\pi}\).

This conversion is necessary when switching between measurements, especially in advanced mathematics or physics, where radians are preferred for their mathematical properties.

Simplifying Fractions

Simplifying fractions plays an important role when you're converting degrees to radians. After using the conversion formula, you often end up with a fraction that can be reduced to its simplest form. This step ensures the result is as clear and concise as possible. Here's a quick reminder on how to simplify:

• Identify the greatest common divisor (GCD) between the numerator and the denominator.
• Divide both the numerator and the denominator by this GCD to get the simplified fraction.

In our example, after converting \(660^{\circ}\) to radians, the resulting fraction \(\frac{660\pi}{180}\) is simplified to \(\frac{11\pi}{3}\) by dividing both numbers by their GCD, 60. Simplification makes your results more manageable and easier to use in further calculations.