Problem 80 - Algebra and Trigonometry Real Mathematics, Real People

Question

Convert the angle measure from radians to degrees. Round your answer to three decimal places. $$-4.2 \pi$$

Step-by-Step Solution

Answer

The converted angle measure is -756 degrees.

Step 1: Understand the Conversion Factor

Given that \( \pi \) radians is equal to 180 degrees, the conversion factor from radians to degrees is \( \frac{180}{\pi} \) degrees per radian. This will be used in the next step.

Step 2: Apply the Conversion Factor

Multiply the given radian measure of \(-4.2 \pi\) by the conversion factor: \(-4.2 \pi \times \frac{180}{\pi}\). Notice that the \(\pi\) in the numerator and in the denominator will cancel each other out. Now, proceed with the multiplication.

Step 3: Compute the Result

Carrying out the multiplication: \(-4.2 \times 180 = -756\). Therefore the result is that \(-4.2 \pi\) radians is equal to -756 degrees.

Key Concepts

Angle Measurement

Angles can be measured in two primary units: radians and degrees. These units help us express and understand the size of an angle in a geometric figure. Radians are often used in mathematics, especially in calculus, due to their natural connection with the unit circle. One radian is defined as the angle created when the radius of a circle is wrapped along its circumference. On the other hand, degrees are more commonly used in everyday contexts and are based on dividing a circle into 360 equal parts. An understanding of both units is crucial as different contexts and fields prefer different units.

Conversion Factor

When converting between radians and degrees, we use a conversion factor. This factor is essential because it provides the tool we need to switch between these two units. The conversion factor from radians to degrees is derived from the fact that a full circle is 360 degrees and also equals to \(2\pi\) radians. Therefore, 1 radian is equivalent to \(\frac{180}{\pi}\) degrees. This conversion factor allows us to multiply any radian measure by \(\frac{180}{\pi}\) to find its degree equivalent. Knowing this makes it easy to navigate between different mathematical problems involving angle measurements.

Degree Calculation

Degree calculation in conversions involves applying the conversion factor to a given radian measure. For example, to convert \(-4.2\pi\) radians, we utilize the conversion factor \(\frac{180}{\pi}\). By multiplying \(-4.2\pi\) by \(\frac{180}{\pi}\), the \(\pi\) units cancel out due to division and multiplication rules. This leaves us multiplying \(-4.2\) directly by 180. Completing this calculation gives us the degree measure of the original angle in radians, enabling us to understand or use the angle in contexts where degrees are preferred.

Mathematical Multiplication

Mathematical multiplication is a basic but crucial operation when converting angle measurements. For a conversion like \(-4.2\pi\) radians to degrees, the multiplication process is straightforward. We start with the known radian value, then multiply it by the conversion factor, \(\frac{180}{\pi}\). In practice, this action reduces the process to multiplying \(-4.2\) by 180 after cancelling \(\pi\) from the equation, a vital realization when simplifying expressions. This multiplication step, often assisted by calculating tools, helps in different scenarios from academic exercises to real-world applications, ensuring the answer is accurate and makes sense in desired contexts.