Problem 10 - Algebra and Trigonometry with Analytic Geometry

Question

Find the exact radian measure of the angle. (a) \(120^{\circ}\) (b) \(-135^{\circ}\) (c) \(210^{\circ}\)

Step-by-Step Solution

Answer

(a) \(\frac{2\pi}{3}\), (b) \(\frac{-3\pi}{4}\), (c) \(\frac{7\pi}{6}\).

Step 1: Understand the Conversion Formula

To convert from degrees to radians, use the formula: \[ \text{Radians} = \left( \frac{\pi}{180} \right) \times \text{Degrees} \]. This formula allows us to convert any angle in degrees to radians, which is the standard unit of angular measurement in mathematics.

Step 2: Convert 120 Degrees to Radians

Applying the formula: \[ \text{Radians} = \left( \frac{\pi}{180} \right) \times 120 \]. Simplify this: \[= \frac{120\pi}{180} = \frac{2\pi}{3} \]. So, the radian measure of \(120^{\circ}\) is \(\frac{2\pi}{3}\).

Step 3: Convert -135 Degrees to Radians

Using the conversion formula: \[ \text{Radians} = \left( \frac{\pi}{180} \right) \times (-135) \]. Simplifying this: \[= \frac{-135\pi}{180} = \frac{-3\pi}{4} \]. Thus, the radian measure of \(-135^{\circ}\) is \(\frac{-3\pi}{4}\).

Step 4: Convert 210 Degrees to Radians

Use the formula again: \[ \text{Radians} = \left( \frac{\pi}{180} \right) \times 210 \]. Simplifying results in: \[= \frac{210\pi}{180} = \frac{7\pi}{6} \]. Therefore, the radian measure of \(210^{\circ}\) is \(\frac{7\pi}{6}\).

Key Concepts

Angle Measurement

When we talk about angles, they are essentially measures of how much one line has rotated about a point, compared to another line. Think of the hands of a clock. The angle measure gives us a value that quantifies this rotation. While we often see angles represented in degrees, especially in everyday contexts, it's not the only way they can be measured.
Angles can also be measured in radians, where one complete rotation (circle) around a point equals 2π radians. While degrees are split into a circle of 360 parts, radians divide the circle into 2π parts. This method arises from the relationship between the radius of a circle and its circumference.
Understanding both degree and radian measurements of angles helps in various fields of mathematics and physics. It’s especially important in trigonometry as it provides a more natural connection to functions like sine and cosine. Having a strong grasp of both systems allows for easier transition between different areas of scientific and technical study.

Degrees to Radians Conversion

Converting degrees to radians is an essential skill in trigonometry and is crucial for those studying calculus and advanced mathematics. The conversion is necessary because radians offer a more direct connection to mathematical formulas.
The conversion formula is simple but important:

• The formula: \[ \text{Radians} = \left( \frac{\pi}{180} \right) \times \text{Degrees} \]

This formula expresses how many radians are equivalent to a certain degree measure by considering that 180 degrees corresponds to \( \pi \) radians. When you multiply the degrees by \( \pi/180 \), you're essentially scaling the degrees to the radian measure.
For example, when converting 120 degrees to radians, you multiply 120 by \( \pi/180 \), giving \( 2\pi/3 \) as the radian measure. This method applies to both positive and negative angles, allowing you to work within different angular ranges.

Trigonometry

Trigonometry is a branch of mathematics that examines the relationships between the sides and angles of triangles. It is fundamental in studying periodic phenomena like waves and oscillations.
The most basic elements of trigonometry are functions like sine, cosine, and tangent. These functions relate the angles of a right triangle to the ratios of its sides. Importantly, trigonometric functions can be defined using radians, which simplifies many mathematical formulas and calculations. This is particularly true when dealing with periodic functions and complex exponential functions.
Radians provide a natural unit for angle measurement in trigonometry. They are used to measure angles in a circular motion, essential for studies involving rotations and waves. As a student of trigonometry, becoming comfortable with using radians can lead to a deeper understanding of this mathematical field and its applications.