What is the unit circle?
The unit circle is a circle with radius 1 and centre (0, 0). We measure an angle θ from the positive x-axis, turning anticlockwise. Where the arm of the angle meets the circle we get a point P.
We define: cos θ = x-coordinate of P, sin θ = y-coordinate of P, tan θ = y ÷ x. Because the radius is 1, these agree with SOH CAH TOA for acute angles, but now they work for any angle, even 200° or −45°.
Every point on the circle obeys x² + y² = 1, so sin²θ + cos²θ = 1.
Exact values: 30°, 45°, 60° and the quadrants
Half of a square (45°-45°-90°) and half of an equilateral triangle (30°-60°-90°) give exact values:
- θ = π/6 (30°): P = (√3/2, 1/2)
- θ = π/4 (45°): P = (√2/2, √2/2)
- θ = π/3 (60°): P = (1/2, √3/2)
- θ = 0, π/2, π, 3π/2: P = (1, 0), (0, 1), (−1, 0), (0, −1)
Radians: one full turn = 2π, so 180° = π. Multiply degrees by π/180 to change to radians.
Signs: in quadrant I all are +; in II only sin is +; in III only tan is +; in IV only cos is +.
Symmetry: mirror angles
The circle is symmetric, so mirror points give new values for free:
- π − θ (mirror in the y-axis): sin(π − θ) = sin θ, cos(π − θ) = −cos θ
- π + θ (half turn): sin(π + θ) = −sin θ, cos(π + θ) = −cos θ, tan(π + θ) = tan θ
- 2π − θ or −θ (mirror in the x-axis): sin(−θ) = −sin θ, cos(−θ) = cos θ
The reference angle is the acute angle between the arm and the x-axis. Find the value for the reference angle, then fix the sign using the quadrant.
Because cos(−θ) = cos θ, cosine is an even function (its graph is symmetric about the y-axis). Because sin(−θ) = −sin θ, sine is odd (its graph is symmetric about the origin). Tangent is odd too.
Periodicity
Adding 2π brings P back to the same point. So sin(θ + 2π) = sin θ and cos(θ + 2π) = cos θ. The smallest such shift is the period: 2π for sin and cos.
For tan, half a turn already works: P and the opposite point give the same y ÷ x, so the period of tan is π.
Use it to simplify big angles: 750° = 2 × 360° + 30°, so sin 750° = sin 30° = 1/2.
Try it
In the 3D, set θ = 45°, then 135°, 225° and 315°. Write down (x, y) each time. You will see the same number √2/2 ≈ 0.71 with different signs. At home: draw a circle of radius 10 cm on graph paper, mark 30° with a protractor and measure x and y. Divide by 10: you get about 0.87 and 0.5.
Key formulas and definitions
- P = (cos θ, sin θ), tan θ = sin θ / cos θ
- sin²θ + cos²θ = 1
- sin(π − θ) = sin θ, cos(π − θ) = −cos θ
- sin(π + θ) = −sin θ, cos(π + θ) = −cos θ, tan(π + θ) = tan θ
- sin(−θ) = −sin θ, cos(−θ) = cos θ
- sin(θ + 2π) = sin θ, cos(θ + 2π) = cos θ, tan(θ + π) = tan θ
- radians = degrees × π / 180
Worked examples
1. Find the coordinates of P on the unit circle at θ = π/3.
π/3 = 60°. From the 30-60-90 triangle, cos 60° = 1/2 and sin 60° = √3/2. So P = (1/2, √3/2).
2. Find sin 150°.
150° = 180° − 30° (quadrant II). Reference angle 30°. sin is positive in II. sin 150° = sin 30° = 1/2.
3. Find cos 225°.
225° = 180° + 45° (quadrant III). cos is negative in III. cos 225° = −cos 45° = −√2/2.
4. Find tan(−π/4).
tan is odd: tan(−π/4) = −tan(π/4) = −1.
5. Find sin(11π/6).
11π/6 = 2π − π/6 (quadrant IV). sin is negative in IV. sin(11π/6) = −sin(π/6) = −1/2.
6. Simplify cos 780° and tan 600°.
780° − 2 × 360° = 60°, so cos 780° = cos 60° = 1/2. tan has period 180°: 600° − 3 × 180° = 60°, so tan 600° = tan 60° = √3.
7. If sin θ = 3/5 and θ is in quadrant II, find cos θ and tan θ.
cos²θ = 1 − 9/25 = 16/25, so cos θ = ±4/5. In II cos is negative: cos θ = −4/5. tan θ = (3/5) ÷ (−4/5) = −3/4.
Common mistakes
- Swapping x and y: cos is the x-coordinate, sin is the y-coordinate.
- Forgetting the sign: the reference angle gives the size, the quadrant gives the sign.
- Thinking the period of tan is 2π. It is π, because the opposite point gives the same ratio.
- Mixing degrees and radians on a calculator: check the mode before you press sin.