Problem 56 - Precalculus : Building Concepts and Connections

Question

Convert each angle from degrees to radians. $$300^{\circ}$$

Step-by-Step Solution

Answer

The angle \(300^{\circ}\) is equivalent to \(\frac{5\pi}{3}\) radians.

Step 1: Write down the given angle in degrees

The angle given in the problem is \(300^{\circ}\).

Step 2: Set up the conversion

Setup a simple conversion expression where you multiply the given angle by the conversion factor \(\frac{\pi}{180}\). The equation becomes, \(300^{\circ} \times \frac{\pi}{180}\).

Step 3: Perform the Calculation

Perform the multiplication to get the angle in radians. It simplifies to \(\frac{300\pi}{180}\). You can simplify this fraction further by cancelling out a common factor of 60 from both numerator and denominator to get \(\frac{5\pi}{3}\).

Key Concepts

Degrees to Radians

Angles are frequently expressed in both degrees and radians. However, mathematical operations are often simpler when using radians. Converting degrees to radians involves using a conversion factor. This factor derives from the relationship that a full circle is both 360 degrees and \(2\pi\) radians. Thus, the conversion factor is \(\frac{\pi}{180}\).To convert an angle from degrees to radians:

• First, write down the angle you need to convert. For example, \(300^{\circ}\).
• Next, multiply these degrees by the conversion factor \(\frac{\pi}{180}\).
• Finally, simplify the resulting fraction if needed.

This process allows us to express the angle in a form suitable for most mathematical calculations, especially those involving trigonometry.

Mathematical Conversion

Understanding conversions is essential in mathematics, whether it's converting units of length, area, or angles like in this exercise. A mathematical conversion involves taking a quantity expressed in one unit and re-expressing it in another unit through a precise mathematical operation.For angles, degrees are often converted into radians to streamline calculations in mathematics and physics. The formula for conversion is derived from the equivalence of 360 degrees to \(2\pi\) radians. Therefore:

• The conversion factor \(\frac{\pi}{180}\) is derived because \(\frac{2\pi}{360} = \frac{\pi}{180}\).
• Multiply the degree measure by this factor.
• For example, \(300^{\circ} \times \frac{\pi}{180} = \frac{300\pi}{180}\).

Simplifying this fraction often gives us a much clearer radian measure.

Trigonometry

Trigonometry is the branch of mathematics that deals with the study of triangles, particularly right-angled triangles, and their properties. A crucial aspect of trigonometry is angle measures, where radians provide a more natural measurement system than degrees. The reason behind using radians in trigonometry is its deep connection with the unit circle and calculus. When an angle is measured in radians:

• The arc length of a circle segment can be directly calculated with simple trigonometric identities.
• Key trigonometric functions, such as sine and cosine, have derivatives that are simplified when the angle is in radians.
• This makes solving equations involving trigonometric identities much more straightforward.

Angles measured in radians make formulas and theorems in trigonometry cleaner and particularly useful in higher mathematical contexts.