Understanding Radians 📐

Radians are a way to measure angles, just like degrees. They are widely used in mathematics, physics, and engineering because of their natural relationship with circles and trigonometric functions.


What is a Radian? 🤔

A radian is the angle created when the length of an arc on a circle is equal to the radius of that circle.

Key Facts:


Why Use Radians? 🧮

Radians simplify the relationship between an angle and the arc length of a circle. For example:

This makes radians the natural choice for mathematical calculations involving circles.


Converting Between Degrees and Radians 🔄

To convert degrees to radians:

Multiply the angle in degrees by π/180.

To convert radians to degrees:

Multiply the angle in radians by 180/π.

Examples:

  1. Convert 90 degrees to radians:

    • 90 × π / 180 = π/2 radians
    • Approximately 1.57 radians.
  2. Convert π radians to degrees:

    • π × 180 / π = 180 degrees.

How Radians Relate to the Unit Circle 🎯

In the unit circle:


Practice Problems 📝

Problem 1:

Convert the following angles to radians:

  1. 45 degrees
  2. 120 degrees
  3. 210 degrees

Problem 2:

Convert the following radians to degrees:

  1. π/6
  2. 5π/4

Problem 3:

A wheel with a radius of 10 cm rolls 20 cm along a flat surface. What angle (in radians) does the wheel rotate through?