The Unit Circle - Trigonometry Beyond Right Triangles
The unit circle is a circle with a radius of exactly 1 unit, centred at the origin of a coordinate plane. It is the tool that extends trigonometric ratios beyond right triangles and acute angles to any angle – including negative angles and angles greater than 360°.
Why the Unit Circle?
In a right triangle, sine and cosine are only defined for angles between 0° and 90°. The unit circle breaks that restriction. For any angle θ, the point where the terminal side meets the unit circle is ( cos(θ), sin(θ) ). This gives sine and cosine values for every possible angle.
Reading the Unit Circle
| Angle | Radians | cos(θ) | sin(θ) | tan(θ) |
|---|---|---|---|---|
| 0° | 0 | 1 | 0 | 0 |
| 30° | π/6 | √3/2 | 1/2 | 1/√3 |
| 45° | π/4 | √2/2 | √2/2 | 1 |
| 60° | π/3 | 1/2 | √3/2 | √3 |
| 90° | π/2 | 0 | 1 | undefined |
| 180° | π | −1 | 0 | 0 |
| 270° | 3π/2 | 0 | −1 | undefined |
| 360° | 2π | 1 | 0 | 0 |
Degrees and Radians
Angles on the unit circle are often measured in radians rather than degrees.
360° = 2π radians so 1 radian ≈ 57.3°
To convert: degrees × (π / 180) = radians.
Radians are the natural unit for calculus and advanced mathematics.
The Four Quadrants
The coordinate plane is divided into four quadrants. The signs of sine and cosine change across these quadrants:
Quadrant I (0°–90°): sin > 0, cos > 0, tan > 0
Quadrant II (90°–180°): sin > 0, cos < 0, tan < 0
Quadrant III (180°–270°): sin < 0, cos < 0, tan > 0
Quadrant IV (270°–360°): sin < 0, cos > 0, tan < 0
Memory aid: All Students Take Calculus – All, Sin, Tan, Cos positive in quadrants I, II, III, IV respectively.
Reference Angles
The reference angle is the acute angle formed between the terminal side of θ and the nearest x-axis.
It lets you find trig values for any angle using the same table as 0°–90°, then adjust the sign for the correct quadrant.
Worked Examples
150° is in Quadrant II. Reference angle = 180° − 150° = 30°.
sin(150°) = sin(30°) = 1/2 (positive in QII).
cos(150°) = −cos(30°) = −√3/2 (negative in QII).
270 × (π/180) = 270π/180 = 3π/2 radians.
Key Takeaways
- On the unit circle, a point at angle θ has coordinates (cos(θ), sin(θ)).
- 360° = 2π radians. Multiply degrees by π/180 to convert.
- Signs: All positive Q1, Sin positive Q2, Tan positive Q3, Cos positive Q4.
- Use the reference angle + sign rule to find trig values for any angle.
Practice Questions
- State the coordinates of the point on the unit circle at 0°, 90°, 180° and 270°.
- Convert 120° to radians.
- Find sin(240°) and cos(240°) using a reference angle.
- In which quadrant does 315° lie? What is its reference angle?
- Find tan(135°) using the unit circle.