Problem 57 - Algebra 2

Question

Rewrite each degree measure in radians and each radian measure in degrees. (lesson \(13-2 )\) \(\frac{5 \pi}{3}\)

Step-by-Step Solution

Answer

300 degrees.

Step 1: Identify the conversion formula

To convert from radians to degrees, use the formula: \[ \text{Degrees} = \text{Radians} \times \frac{180}{\pi} \] Here, we need to convert \( \frac{5\pi}{3} \) radians to degrees.

Step 2: Substitute the radians into the formula

We substitute \( \frac{5\pi}{3} \) for Radians in the conversion formula:\[ \text{Degrees} = \frac{5\pi}{3} \times \frac{180}{\pi} \]

Step 3: Simplify the expression

Cancel out \( \pi \) in the numerator and denominator to simplify:\[ \text{Degrees} = \frac{5}{3} \times 180 \]

Step 4: Perform the multiplication

Multiply \( \frac{5}{3} \) by 180:\[ \text{Degrees} = 5 \times 60 = 300 \]

Step 5: Write the final answer

The measure \( \frac{5\pi}{3} \) radians is equivalent to 300 degrees.

Key Concepts

Radians to Degrees

When working with angles, we often need to convert between radians and degrees. This is because different situations and problems may require one unit over the other. Radians and degrees are two common units used for measuring angles.

• Degrees are a more intuitive and traditional way to measure angles, often taught in early mathematics.
• Radians are used more frequently in higher level mathematics and physics due to their natural relationship with circles.

Understanding how to convert between these units is essential. If we take the mathematical circle, a full circle is 360 degrees or in terms of radians, it's equal to \(2\pi\) radians. This relationship forms the basis of our conversion between radians and degrees.

Conversion Formula

To convert any measurement from radians to degrees, we utilize a simple conversion formula derived from the relationship of one full circle:\[\text{Degrees} = \text{Radians} \times \frac{180}{\pi} \]
This formula emerges from the fact that 360 degrees equals \(2\pi\) radians. Therefore, to find how many degrees correspond to one radian, we recognize that \(\pi\) radians equal 180 degrees. This is why we multiply the radian measure by \(\frac{180}{\pi}\).
To use this formula, simply substitute the angle in radians; in our example, this is \(\frac{5\pi}{3}\), into the formula and perform the calculation to find the equivalent angle in degrees.

Simplifying Expressions

When using the conversion formula, a key step is to simplify the mathematical expression. Simplifying makes the calculations more manageable and reduces the chance of error. In our scenario with \(\frac{5\pi}{3}\) radians, we took the following steps:1. **Cancellation**: \(\pi\) is present both in the numerator of the radian measure and in the denominator of the conversion fraction \(\frac{180}{\pi}\). Cancelling \(\pi\) in both spots simplifies our expression.2. **Multiplication**: After cancelling \(\pi\), you're left with \(\frac{5}{3} \times 180\). To simplify this expression further, carry out the multiplication step-by-step to get \[ \text{Degrees} = \frac{5 \times 180}{3} = 300 \]3. **Result**: Therefore, the radian measure of \(\frac{5\pi}{3}\) simplifies to 300 degrees.By carefully simplifying expressions, you ensure clarity and accuracy in your calculations. This systematic approach helps you work efficiently with both simple and complex expressions.