Source: Unit 6c Unit Circle Packet, Texas A&M University
Tags: unit circle, special right triangles, 45-45-90, 30-60-90, unit circle coordinates, quadrant coordinates, reference angle, symmetry, unit circle chart
Every point on the unit circle can be written as (cos θ, sin θ). The coordinates at all the key angles come from just two special right triangles: the 45-45-90 and the 30-60-90. Once you know the first-quadrant values, symmetry gives you the rest.
Unit circle
A circle centred at the origin with radius 1. Because r = 1, the coordinates of any point on the circle are (cos θ, sin θ).
Special right triangles
The 45-45-90 triangle (isosceles right triangle) and the 30-60-90 triangle. These produce the exact trigonometric values for the key unit-circle angles.
45-45-90 triangle
Sides in ratio 1 : 1 : √2. When the hypotenuse is 1 (unit circle), each leg is √2/2.
30-60-90 triangle
Sides in ratio 1 : √3 : 2. When the hypotenuse is 1, the short leg (opposite 30°) is 1/2 and the long leg (opposite 60°) is √3/2.
Reference angle
The acute angle between the terminal ray and the x-axis. It determines the magnitude of the coordinates; the quadrant determines the signs.
Quadrant sign rules (ASTC)
A mnemonic for which trig functions are positive in each quadrant: All (I), Sine (II), Tangent (III), Cosine (IV). Sometimes remembered as "All Students Take Calculus."
These are the four points where the unit circle crosses the axes. No triangle is needed here, just the coordinates of those intersection points.
0 radians (0°): (1, 0)
π/2 radians (90°): (0, 1)
π radians (180°): (-1, 0)
3π/2 radians (270°): (0, -1)
2π radians (360°): back to (1, 0)
Drop a perpendicular from any π/4-family point on the unit circle to the x-axis. You get a 45-45-90 triangle with hypotenuse 1. Both legs equal √2/2 ≈ 0.707.
π/4 (45°): (√2/2, √2/2) — Quadrant I, both positive
3π/4 (135°): (-√2/2, √2/2) — Quadrant II, x negative
5π/4 (225°): (-√2/2, -√2/2) — Quadrant III, both negative
7π/4 (315°): (√2/2, -√2/2) — Quadrant IV, y negative
The pattern: the magnitudes are always √2/2 for both coordinates. Only the signs change by quadrant.
π/6 angles (30° family): the reference angle is 30°, so the triangle has short leg = 1/2 and long leg = √3/2. The short leg is vertical (the y-value) and the long leg is horizontal (the x-value).
π/6 (30°): (√3/2, 1/2)
5π/6 (150°): (-√3/2, 1/2)
7π/6 (210°): (-√3/2, -1/2)
11π/6 (330°): (√3/2, -1/2)
π/3 angles (60° family): the reference angle is 60°, so the short leg is now horizontal (x-value = 1/2) and the long leg is vertical (y-value = √3/2).
π/3 (60°): (1/2, √3/2)
2π/3 (120°): (-1/2, √3/2)
4π/3 (240°): (-1/2, -√3/2)
5π/3 (300°): (1/2, -√3/2)
You only need to memorise the first-quadrant coordinates. For any angle in another quadrant:
Find the reference angle (the acute angle to the nearest x-axis).
Use the same coordinate magnitudes as the first-quadrant version of that reference angle.
Apply the correct signs based on the quadrant.
Quadrant I: (+, +). Quadrant II: (-, +). Quadrant III: (-, -). Quadrant IV: (+, -).
Angle (rad) | Angle (deg) | Coordinates (cos θ, sin θ) |
|---|---|---|
0 | 0° | (1, 0) |
π/6 | 30° | (√3/2, 1/2) |
π/4 | 45° | (√2/2, √2/2) |
π/3 | 60° | (1/2, √3/2) |
π/2 | 90° | (0, 1) |
2π/3 | 120° | (-1/2, √3/2) |
3π/4 | 135° | (-√2/2, √2/2) |
5π/6 | 150° | (-√3/2, 1/2) |
π | 180° | (-1, 0) |
7π/6 | 210° | (-√3/2, -1/2) |
5π/4 | 225° | (-√2/2, -√2/2) |
4π/3 | 240° | (-1/2, -√3/2) |
3π/2 | 270° | (0, -1) |
5π/3 | 300° | (1/2, -√3/2) |
7π/4 | 315° | (√2/2, -√2/2) |
11π/6 | 330° | (√3/2, -1/2) |
2π | 360° | (1, 0) |
45-45-90 triangle (hypotenuse = 1): legs = √2/2 each
30-60-90 triangle (hypotenuse = 1): short leg (opposite 30°) = 1/2, long leg (opposite 60°) = √3/2
Point on the unit circle: P = (cos θ, sin θ)
⚠️ The three coordinate values you need to memorise are 1/2, √2/2, and √3/2. Everything on the unit circle is built from these.
⚠️ A very common mix-up: confusing which coordinate is 1/2 and which is √3/2 at π/6 vs. π/3. Remember: at π/6 (30°), the angle is small, so the point is close to the x-axis, meaning the y-value is the smaller one (1/2) and the x-value is the larger one (√3/2). At π/3, it flips.
⚠️ Signs come from the quadrant, not the angle number. Always determine the quadrant first, then assign +/-.
⚠️ "All Students Take Calculus" (ASTC) tells you which functions are positive: All in QI, Sin/Csc in QII, Tan/Cot in QIII, Cos/Sec in QIV.
Q: What are the coordinates of the point on the unit circle at 5π/4?
A: (-√2/2, -√2/2). The reference angle is π/4, so both magnitudes are √2/2. Quadrant III means both are negative.
Q: At which angle is the unit circle point (−1/2, √3/2)?
A: 2π/3 (120°). The x-coordinate is negative and y is positive, so it's in Quadrant II. The reference angle with those magnitudes (1/2 and √3/2) is π/3, and π - π/3 = 2π/3.
Q: What are the side lengths of a 30-60-90 triangle with hypotenuse 1?
A: The side opposite 30° is 1/2 and the side opposite 60° is √3/2.
Q: In which quadrants is the x-coordinate (cosine) negative?
A: Quadrants II and III (left side of the circle).
Q: What is the coordinate at 11π/6?
A: (√3/2, -1/2). Reference angle is π/6, Quadrant IV, so x is positive and y is negative.
unit circle, unit circle coordinates, unit circle chart, special right triangles, 45-45-90 triangle, 30-60-90 triangle, reference angle, ASTC, All Students Take Calculus, quadrant sign rules, cos θ = x, sin θ = y, unit circle memorisation, √2/2, √3/2, 1/2, unit circle values, AP Calculus unit circle, Texas A&M Calculus Unit 6