Problem 9 - Precalculus: Functions and Graphs

Question

Exer. \(9-12:\) Find the exact radian measure of the angle. (a) \(150^{\circ}\) (b) \(-60^{\circ}\) (c) \(225^{\circ}\)

Step-by-Step Solution

Answer

(a) \(\frac{5\pi}{6}\), (b) \(-\frac{\pi}{3}\), (c) \(\frac{5\pi}{4}\).

Step 1: Understand Degree to Radian Conversion Formula

To convert degrees to radians, we use the formula: \[ \text{Radians} = \frac{\pi}{180^{\circ}} \times \text{Degrees} \]. This formula will help us convert any angle given in degrees into radians.

Step 2: Convert 150 Degrees to Radians

Apply the conversion formula to \(150^{\circ}\): \[ \text{Radians} = \frac{\pi}{180^{\circ}} \times 150^{\circ} \]. Simplifying, \( \frac{{150\pi}}{{180}} = \frac{5\pi}{6} \). Thus, the radian measure of \(150^{\circ}\) is \( \frac{5\pi}{6} \).

Step 3: Convert -60 Degrees to Radians

Apply the conversion formula to \(-60^{\circ}\): \[ \text{Radians} = \frac{\pi}{180^{\circ}} \times (-60^{\circ}) \]. Simplifying, \( \frac{{-60\pi}}{{180}} = -\frac{\pi}{3} \). Thus, the radian measure of \(-60^{\circ}\) is \(-\frac{\pi}{3} \).

Step 4: Convert 225 Degrees to Radians

Apply the conversion formula to \(225^{\circ}\): \[ \text{Radians} = \frac{\pi}{180^{\circ}} \times 225^{\circ} \]. Simplifying, \( \frac{{225\pi}}{{180}} = \frac{5\pi}{4} \). Thus, the radian measure of \(225^{\circ}\) is \( \frac{5\pi}{4} \).

Key Concepts

Degree to Radian Conversion

Converting angles from degrees to radians is a crucial skill in trigonometry and calculus. Let's delve into why it matters and how you can do it. Angles can be measured in two different units: degrees and radians. While degrees are often used in everyday contexts and in initial learning phases, radians provide a natural way to relate angles to circles and arcs, and they are fundamental in advanced mathematics.
To convert degrees to radians, we use a simple conversion rule:

• The full circle in degrees sums up to 360.
• In radians, a full circle is measured as \(2\pi\).
• So, \(180^{\circ}\) is equivalent to \(\pi\) radians.

This leads us to the conversion formula:\[\text{Radians} = \frac{\pi}{180^{\circ}} \times \text{Degrees}\]By using this formula, you can convert any degree measure into radians, making it much easier to tackle problems involving angular motion and periodic functions.

Angle Measurement

Understanding how angles are measured is essential when working with trigonometry. Angles can be defined as the space between two intersecting lines or surfaces at or close to the point where they meet. They can be quantified in various units:

• Degrees: This is the most common and easily understandable unit. A degree is 1/360 of a full rotation.
• Radians: The radian is another way to measure angles, primarily used in mathematics. One radian is the angle created when the arc length is equal to the radius of the circle.

A radian is naturally related to the geometry of circles, which makes it a favorable unit in mathematics. Understanding these units can vastly improve how you interpret problems in calculus and physics, especially those involving circular motion and wave dynamics.

Trigonometry Basics

Trigonometry forms the bridge between geometry and algebra and is foundational for understanding wider concepts in mathematics. It primarily deals with relationships in triangles, specifically right-angle triangles. Here are some foundational concepts:

• Sine, Cosine, and Tangent: These are basic trigonometric functions that relate the angles of a triangle to the lengths of its sides.
• Pythagorean Theorem: Relates the square of the hypotenuse of a right triangle to the sum of the squares of the other two sides.
• Unit Circle: A circle with a radius of one, centered at the origin of a coordinate system. It simplifies the understanding of trigonometric functions and identities.

By getting familiar with these basics, you can solve problems involving angles and understand how they apply across different contexts, such as physics and engineering.