Problem 64 - Algebra and Trigonometry Real Mathematics, Real People

Question

Rewrite each angle in radian measure as a multiple of \(\pi .\) (Do not use a calculator.) (a) \(315^{\circ}\) (b) \(120^{\circ}\)

Step-by-Step Solution

Answer

\(315^{\circ} = \frac{7\pi}{4}\) radian and \(120^{\circ} = \frac{2\pi}{3}\) radian.

Step 1: Convert 315 degrees to radian

For the first conversion, use the proportion: \(1^{\circ} = \frac{2\pi}{360}\) radian. So \(315^{\circ} = 315 \cdot \frac{2\pi}{360} = \frac{7\pi}{4}\) radian.

Step 2: Convert 120 degrees to radian

Similarly, for the second conversion: \(1^{\circ} = \frac{2\pi}{360}\) radian. Therefore \(120^{\circ} = 120 \cdot \frac{2\pi}{360} = \frac{2\pi}{3}\) radian.

Key Concepts

Angle Measure Conversion

In the world of geometry and trigonometry, angles can be measured in different units. The most common units are degrees and radians. Understanding how to convert between these units is essential for solving various mathematical problems. The fundamental relationship for converting from degrees to radians is based on the equation \(1^{\circ} = \frac{\pi}{180} \text{ radians}\). This means every 1 degree is equivalent to \(\frac{\pi}{180}\) of a radian.

To convert an angle in degrees to radians, you multiply the degree measure by \(\frac{\pi}{180}\). For example, to convert \(315^{\circ}\) to radians, you calculate:

• Multiply 315 by \(\frac{\pi}{180}\) to get \(315 \cdot \frac{\pi}{180} = \frac{7\pi}{4}\) radians.

This formula helps by neatly cancelling out the degree units, leaving you with a pure radian measure.

Similarly, for \(120^{\circ}\), apply the same conversion:

• Multiply 120 by \(\frac{\pi}{180}\) to get \(120 \cdot \frac{\pi}{180} = \frac{2\pi}{3}\) radians.

By consistently using this method, you can seamlessly switch between degrees and radians for any angle.

Radian Measure

Radians are a way to measure angles that relate directly to the circle's radius. Unlike degrees, which are arbitrary and based on the ancient Babylonian base-60 system, radians are a natural part of circle geometry. One complete revolution around a circle is \(2\pi\) radians, equivalent to 360 degrees. Therefore, a circle has exactly \(2\pi\) radians around it.

The radian measure is based on the arc length of a circle and its radius. A radian is defined as the angle created when the arc length is equal to the circle's radius. Due to this relationship, radians provide a straightforward method for calculating real-world applications in physics and engineering.

Another important note is that angles written as multiples of \(\pi\) (like \(\frac{7\pi}{4}\) for \(315^{\circ}\) or \(\frac{2\pi}{3}\) for \(120^{\circ}\)) are considered simplified and standardized, making mathematical expressions neater and more universally understood.

Trigonometry

Trigonometry is a branch of mathematics that studies relationships involving lengths and angles of triangles. It is a fundamental aspect of geometry based on these angles, especially as they increase and decrease cyclically when a circle is involved. Understanding how to measure these angles in radians is crucial in trigonometry.

Trigonometric functions such as sine, cosine, and tangent rely heavily on radian measure because of their periodic nature. The unit circle, which is central to trigonometry, is defined in terms of radians, making it easier to understand and compute trigonometric functions for angles that might otherwise seem complex in degree measure. When stated in radians, these functions become manageable as sine and cosine values repeat every \(2\pi\) radians, and tangent repeats every \(\pi\) radians.

Overall, the synergy between trigonometry and radians enhances the simplicity and elegance of solving complex problems in various fields, from basic geometry to advanced engineering.