Understanding Radians

Definition

• A Radian is a unit of angle measure.
• 1 Radian ≈ 57.3 degrees.

How is 1 Radian Derived?

• A full circle is 360 degrees or 2π radians.
• Formula:
  • 360 degrees = 2π radians.
  • Therefore, 1 radian = 360 / (2π) degrees ≈ 57.29577951 degrees (rounded to 57.3 degrees).

Conceptualizing Radians

• Radius of a circle: Distance from the center to any point on the circle.
• Arc Length (s): Distance along the circle between two points.
• Radian occurs when the arc length equals the radius of the circle.
  • Formula: s = θ × r
  • When s = r, θ (the angle) = 1 radian.

Units of Measurement

• θ (Theta) = Angle in radians.
• r = Radius of the circle.
• s = Arc length.
• For example, with a radius of 4 inches and an angle of 2 radians, the arc length is 8 inches.

How Many Radians in a Circle?

• Circle circumference = 2πr.
• Full circle in radians = 2π radians ≈ 6.283 radians.

Converting Between Radians and Degrees

Degrees to Radians

• Multiply degrees by π/180.
• Example: Convert 30 degrees to radians.
  • 30 * (π/180) = (30π/180) = π/6 radians.
• Example: Convert 60 degrees to radians.
  • 60 * (π/180) = (60π/180) = π/3 radians.

Radians to Degrees

• Multiply radians by 180/π.
• Example: Convert 5π/6 radians to degrees.
  • (5π/6) * (180/π) = 150 degrees.
• Example: Convert 4π/3 radians to degrees.
  • (4π/3) * (180/π) = 240 degrees.

Summary

• Radians measure angles based on the radius length.
• Conversion between radians and degrees is essential for calculations in geometry and trigonometry.