Problem 59 - Precalculus : Building Concepts and Connections

Question

Convert each angle from degrees to radians. $$720^{\circ}$$

Step-by-Step Solution

Answer

The angle \(720^{\circ}\) in radians is \(4π\) radians.

Step 1: Identifying the angle in degrees

The given angle measure is \(720^{\circ}\).

Step 2: Using the conversion factor

To convert degrees to radians, multiply the given degree measurement by \(π/180\). Therefore we have the following equation: \(720^{\circ} * π/180\).

Step 3: Carrying out the calculation

Multiplying out gives \(4π\) radians.

Key Concepts

Converting Degrees to Radians

Angles can be expressed either in degrees or radians. In many mathematical applications, it's necessary to convert between these two units. Degrees are more common in everyday measurements, such as in navigation or when describing geometric figures. However, radians are predominantly used in calculus and trigonometry because they relate directly to the properties of circles and are the standard unit of angular measure in many mathematical contexts.

Understanding how to convert from degrees to radians is essential. The key is recognizing that one full circle is equal to both 360 degrees and \(2\pi\) radians. Thus, there is a direct relationship between these two measures that allows for conversion.

Understanding the Conversion Factor

To convert degrees to radians, a conversion factor is necessary. This factor arises from the relationship of degrees to radians in a full circle as outlined previously. Since a full circle is \(360^{\circ}\) and also equals \(2\pi\) radians, the conversion factor becomes clear.

Specifically, for every full circle (or \(360^{\circ}\)), there are \(2\pi\) radians. Simplifying this relationship gives us the conversion factor:

• \(\pi/180\) radians per degree.

This factor lets us easily transform degree measures into radians by multiplication. Remembering this factor, \(\pi/180\), allows one to quickly make necessary conversions in mathematical problems.

Radians Calculation Step-by-Step

Computing radians from degrees is straightforward using the conversion factor. Let’s look at the calculation, as demonstrated in the given problem, where \(720^{\circ}\) needed to be converted to radians.

The process involves taking the degree measure and multiplying it by \(\pi/180\). Here's how it works step-by-step with \(720^{\circ}\):

• Start with the degree measurement: \(720^{\circ}\).
• Apply the conversion factor: multiply \(720\) by \(\pi/180\).
• Perform the multiplication: \(720 \times \pi/180 = 4\pi\) radians.

The result is \(4\pi\) radians. By carrying out these steps, any angle can be easily converted from degrees to radians using the consistent method of multiplying by \(\pi/180\). This ensures accuracy and simplicity in various mathematical and scientific applications.