This topic focuses on defining the trigonometric functions cosθ, sinθ, and tanθ using the unit circle.
Understanding trigonometric graphs, identities, and the ambiguous case of the sine rule is clearer through geometric interpretation on the unit circle.
| Concept | Meaning / Explanation |
|---|---|
| Unit circle | A circle centred at the origin with radius 1. Any point on the unit circle has coordinates (cosθ, sinθ), where θ is the angle from the positive x-axis. |
| sinθ and cosθ | For a point P on the unit circle:
x-coordinate = cosθ y-coordinate = sinθ |
| Pythagorean identity | On the unit circle: cos2θ + sin2θ = 1 Derived from the equation of the unit circle: x2 + y2 = 1. |
| tanθ | tanθ = sinθ ÷ cosθ Represented on the unit circle as the slope of the terminal ray. |
| Ambiguous case (sine rule) | Occurs in non-right triangles when two sides and a non-included angle (SSA) are given. Sometimes two possible triangles exist, one, or none. Understanding the unit circle helps interpret why sinθ repeats values. |
| Graphical solutions | Trigonometric equations can be solved by graphing y = f(x) on a finite interval and identifying intersection points. |
📌 1. Understanding trigonometric functions from the unit circle
The unit circle provides a unified, geometric definition for sinθ, cosθ, and tanθ:
- Each angle θ corresponds to a point P(cosθ, sinθ).
- This point moves smoothly as θ increases, which naturally produces the wave-like graphs of sine and cosine.
- sinθ corresponds to vertical projection; cosθ corresponds to horizontal projection.
- tanθ is the gradient (rise ÷ run) of the radius line.
Because the unit circle wraps around itself, values of sinθ and cosθ repeat every 2π, explaining periodicity.
📌 2. Worked example — interpreting the unit circle
Example
For θ = 150°, find cosθ, sinθ and tanθ using the unit circle interpretation.
Step 1 — Convert to radians
150° = 150 × (π ÷ 180) = 5π ÷ 6
Step 2 — Determine coordinates
Angle 150° lies in quadrant II, where cosine is negative and sine is positive.
Coordinates on the unit circle: (−√3 ÷ 2, 1 ÷ 2)
Step 3 — Calculate tanθ
tanθ = sinθ ÷ cosθ = (1 ÷ 2) ÷ (−√3 ÷ 2) = −1 ÷ √3
Thus:
cosθ = −√3 ÷ 2
sinθ = 1 ÷ 2
tanθ = −1 ÷ √3
🧮 GDC Use
- Graph sinx, cosx and tanx to visualise intersections when solving equations.
- Use trace mode to identify exact or approximate values of θ where the graph equals a given number.
- Check ambiguous case solutions by graphing sinθ to see why two possible values of θ yield the same sine.
- Ensure the GDC is in radian mode for AHL topics unless specifically using degrees.
🌍 Real-World Applications
- Electrical engineering uses sinusoidal voltage and current, which follow the sine function exactly.
- Mechanical systems (rotors, gears, oscillators) use unit-circle-based rotational models.
- Computer graphics, wave simulation and signal processing use cosine-sine decomposition.
📐 IA Spotlight
- Investigate how the unit circle leads to periodic motion in mechanics or electricity.
- Model a real-life oscillation (pendulum, spring, alternating current) using sine or cosine functions.
- Explore the ambiguous case through geometric construction and compare exact vs approximate methods.
🔍 TOK Perspective
- The word “sine” has cultural roots in Indian and Arabic mathematics—does this show that mathematics is not culturally neutral?
- Unit-circle trigonometry evolved across multiple civilizations; how does mathematical knowledge develop through intercultural exchange?
- Many identities (e.g., cos2θ + sin2θ = 1) arise from geometry. Does this suggest mathematics is discovered rather than invented?
📝 Paper Tips
- Sketch the unit circle whenever possible—visualising quadrants helps avoid sign errors.
- Use radians unless the question specifically mentions degrees.
- For solving equations like sinx = a, always consider solutions in the full interval (e.g. 0 ≤ x ≤ 2π).
- For ambiguous case questions, consider the geometry—not just the algebraic sine rule.