Problem 10 - Precalculus: Functions and Graphs

Question

Exer. \(9-12:\) Find the exact radian measure of the angle. (a) \(120^{\circ}\) (b)-135^{0}\( (c) \)210^{\circ}$

Step-by-Step Solution

Answer

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

Step 1: Understanding the Formula

To convert degrees to radians, use the formula: \( \text{radians} = \text{degrees} \times \frac{\pi}{180} \). This formula is derived from the ratio of degrees to radians in a circle, which is 360 degrees to \(2\pi\) radians.

Step 2: Convert 120 degrees to radians

Using the formula, convert \(120^{\circ}\) to radians: \(120 \times \frac{\pi}{180} = \frac{120\pi}{180} = \frac{2\pi}{3}\). Thus, the radian measure for \(120^{\circ}\) is \(\frac{2\pi}{3}\).

Step 3: Convert -135 degrees to radians

Using the conversion formula, change \(-135^{\circ}\) to radians: \(-135 \times \frac{\pi}{180} = -\frac{135\pi}{180} = -\frac{3\pi}{4}\). The radian measure for \(-135^{\circ}\) is \(-\frac{3\pi}{4}\).

Step 4: Convert 210 degrees to radians

Apply the formula to convert \(210^{\circ}\) to radians: \(210 \times \frac{\pi}{180} = \frac{210\pi}{180} = \frac{7\pi}{6}\). Therefore, the radian measure for \(210^{\circ}\) is \(\frac{7\pi}{6}\).

Key Concepts

Degree to Radian Conversion

Understanding the conversion from degrees to radians is a key skill in many areas of mathematics, especially in trigonometry. This conversion uses the formula:\[ \text{radians} = \text{degrees} \times \frac{\pi}{180} \]This formula originates from the fact that a full circle is 360 degrees, which is equivalent to \(2\pi\) radians.

• To convert, multiply the degree measure by \(\frac{\pi}{180}\).
• The factor \(\frac{\pi}{180}\) represents the relationship between degrees and radians as a ratio.
• This helps to translate angles from a degree-based system to a radian-based system, which is often required in calculus and higher-level applications.

For instance, converting \(120^{\circ}\), we multiply \(120 \times \frac{\pi}{180}\), which simplifies to \(\frac{2\pi}{3}\) radians. Similarly, an angle of \(-135^{\circ}\) converts to \(-\frac{3\pi}{4}\) radians and \(210^{\circ}\) becomes \(\frac{7\pi}{6}\) radians.
Understanding this conversion is fundamental for analyzing angles beyond basic triangle problems, especially when dealing with periodic functions in mathematics.

Angle Measurement

Angles are a fundamental component of geometry and trigonometry. They can be measured in two common units: degrees and radians.

• Degrees: A degree is 1/360th of a full rotation around a point. This is the more intuitive system for most everyday situations, like navigation and simple geometry.
• Radians: A radian is the measure of an angle obtained by taking the radius of a circle and wrapping it along the circle's edge. Since there are \(2\pi\) radians in a full circle, this system is especially useful in higher mathematics.

Unlike degrees, radians are more naturally integrated within calculus and the unit circle concept. When working with calculus and trigonometric functions, radian measures often simplify the math involved and provide direct insight into the behavior of functions, especially periodic ones.
Radians help to seamlessly transition between angle measures and the length of an arc on a unit circle, highlighting the intrinsic geometric relationships.

Trigonometry

Trigonometry is the branch of mathematics that explores the relationships between the sides and angles of triangles. It often involves the study of circular functions.

• Key functions include sine, cosine, and tangent, which relate the angles of a triangle to the ratios of its sides.
• The unit circle is a fundamental concept in trigonometry, used to define these functions through radian measures.
• A thorough understanding of degree and radian measures enhances problem-solving skills in trigonometry.

For trigonometry, radians play an important role as they provide a natural way to express angles. When angles are in radians, calculations involving the trigonometric functions like sine and cosine become more straightforward.
This is particularly evident when analyzing waves, oscillations, and other periodic phenomena—common contexts in physics and engineering. Building a strong trigonometric foundation is essential for students aiming to pursue further studies in science, technology, engineering, and mathematics (STEM) fields.