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:

• 1 radian is approximately 57.3 degrees.
• A full circle has 2π radians, which is equivalent to 360 degrees.
• Half a circle (a straight angle) is π radians or 180 degrees.

Why Use Radians? 🧮

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

• The arc length is calculated as:
  • Arc length = Radius × Angle in radians

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:

• The radius is 1.
• The circumference is 2π, so a full rotation equals 2π radians.
• Some common angles and their radian measures:
  • 0 degrees = 0 radians
  • 90 degrees = π/2 radians (about 1.57 radians)
  • 180 degrees = π radians (about 3.14 radians)
  • 270 degrees = 3π/2 radians (about 4.71 radians)
  • 360 degrees = 2π radians (about 6.28 radians)

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
3. 2π

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?