Radians provide a natural mathematical way to measure angles based on the geometry of a circle itself.
This topic extends SL knowledge of circles and connects directly to trigonometric functions, calculus, and periodic modelling.
| Concept | Meaning / Formula |
|---|---|
| Radian | One radian is the angle formed when the arc length equals the radius.
In symbols: arc length s = r gives angle = 1 radian. |
| Conversion | 180° = π radians
1 radian = 180° ÷ π Degrees to radians: θ (radians) = θ° × (π ÷ 180) Radians to degrees: θ° = θ × (180 ÷ π) |
| Arc length | s = rθ (θ in radians only) |
| Area of a sector | A = (1 ÷ 2) × r2 × θ (θ in radians only) |
📌 1. Why radians are the natural unit for angles
- Radians relate angle size directly to the arc length on a circle.
- Many formulas in trigonometry and calculus only work cleanly in radians.
- The radian measure removes arbitrary choices: degrees are human-made, but radians arise naturally from geometry.
- When modelling real-life periodic behaviour (waves, oscillations), radians simplify derivatives and equations.
📌 2. Worked example — converting and applying radian measure
Example
A circle has radius 7 cm. Find:
1. The arc length for an angle of 60°
2. The area of the corresponding sector.
Step 1 — Convert angle to radians
θ = 60° × (π ÷ 180) = π ÷ 3 radians
Step 2 — Arc length
s = rθ = 7 × (π ÷ 3)
s = 7π ÷ 3 cm
Step 3 — Area of sector
A = (1 ÷ 2) × r2 × θ
A = (1 ÷ 2) × 49 × (π ÷ 3)
A = 49π ÷ 6 cm²
🧮 GDC Use
- Use “angle conversion” functions to switch between radians and degrees quickly.
- Graphing radian-based trigonometric functions is more accurate; ensure the calculator mode is set to radians when needed.
- For arc length and sector area, enter formulas directly to check working and avoid arithmetic errors.
🌍 Real-World Applications
- Circular motion in physics uses radians to measure angular displacement, velocity and acceleration.
- Wave motion and diffraction patterns rely on radian measure due to their trigonometric foundations.
- Engineering fields such as robotics and mechanical systems measure rotation using radians for precision.
📐 IA Spotlight
- Explore arc lengths or sector areas in real objects (gears, wheels, tracks) and analyse how angle measurement affects calculations.
- Investigate circular motion and model angular velocity or acceleration using radian-based trigonometric functions.
- Analyse why radians simplify calculus and compare results obtained using degrees vs radians.
🔍 TOK Perspective
- Radians arise from geometry, while degrees are human conventions. Which system provides more “mathematical truth”?
- Is a natural measure (rad) “better” than a human-made one (deg)? By what criteria do mathematicians decide?
- Different cultures used different angle systems—Babylonians used 360 degrees. Does the choice of unit affect how mathematics is understood?
📝 Paper Tips
- Check calculator mode — most mistakes occur from using degree mode in radian questions.
- Sector formulas only work when θ is in radians.
- Leave answers as exact multiples of π unless the question asks for decimals.
- Show conversion steps clearly to earn method marks.