Radian vs. Degree

Algebra-2

1. Fundamental Concepts

Definition: Degrees and radians are two units used to measure angles.
Degrees: A full circle is divided into 360 degrees.
Radians: One radian is the angle subtended at the center of a circle by an arc that is equal in length to the radius of the circle.
Conversion: The relationship between degrees and radians is given by the formula $$\text{{radians}} = \frac{\pi}{180} \cdot \text{{degrees}}$$.

2. Key Concepts

Basic Rule: $$\text{{radians}} = \frac{\pi}{180} \cdot \text{{degrees}}$$

Degree Preservation: The highest degree in the result matches input

Application: Used in trigonometry, physics, and engineering for precise angle measurements

3. Examples

Example 1 (Basic)

Problem: Convert 90 degrees to radians.

Step-by-Step Solution:

Use the conversion formula: $$\text{{radians}} = \frac{\pi}{180} \cdot \text{{degrees}}$$
Substitute 90 for degrees: $$\text{{radians}} = \frac{\pi}{180} \cdot 90$$
Simplify: $$\text{{radians}} = \frac{\pi}{2}$$

Validation: Substitute back into the formula → Original: 90 degrees; Simplified: $$\frac{\pi}{2}$$ radians ✓

Example 2 (Intermediate)

Problem: Convert $$\frac{5\pi}{6}$$ radians to degrees.

Step-by-Step Solution:

Use the inverse conversion formula: $$\text{{degrees}} = \frac{180}{\pi} \cdot \text{{radians}}$$
Substitute $$\frac{5\pi}{6}$$ for radians: $$\text{{degrees}} = \frac{180}{\pi} \cdot \frac{5\pi}{6}$$
Simplify: $$\text{{degrees}} = 150$$

Validation: Substitute back into the formula → Original: $$\frac{5\pi}{6}$$ radians; Simplified: 150 degrees ✓

4. Problem-Solving Techniques

Visual Strategy: Use a unit circle to visualize the relationship between degrees and radians.
Error-Proofing: Always double-check the conversion formulas and ensure the correct values are substituted.
Concept Reinforcement: Practice converting between degrees and radians with various angles to reinforce understanding.