Unit Circle Calculator

How to Use This Calculator

1. Enter an angle in degrees to see its unit circle coordinates (cos θ, sin θ).
2. Use Common Angles to select a standard angle and get exact values.
3. Use Reverse to find the angle from x,y coordinates.

Formula

Example

Frequently Asked Questions

• What is the unit circle? ▼
  The unit circle is a circle with radius 1 centered at the origin of a coordinate system. For any angle θ (measured counterclockwise from the positive x-axis), the terminal point on the circle is (cos θ, sin θ). This definition extends sine and cosine to all real angles — positive, negative, and beyond 360°. The equation of the unit circle is x² + y² = 1, which is the Pythagorean identity in disguise: cos²θ + sin²θ = 1. The unit circle connects circular geometry with trigonometry and is fundamental to understanding periodic functions.
• What are the exact values at 30°, 45°, and 60°? ▼
  The three most important angles and their unit circle coordinates: 30° (π/6): (cos, sin) = (√3/2, 1/2) ≈ (0.866, 0.5). 45° (π/4): (√2/2, √2/2) ≈ (0.707, 0.707). 60° (π/3): (1/2, √3/2) ≈ (0.5, 0.866). These come from the 30-60-90 triangle (sides 1, √3, 2) and the 45-45-90 triangle (sides 1, 1, √2). The pattern: as angle increases from 30° to 60°, cos decreases (√3/2 → 1/2) while sin increases (1/2 → √3/2) — symmetry about 45°.
• What is the ASTC rule? ▼
  The ASTC rule (remembered as "All Students Take Calculus") tells you which trig functions are positive in each quadrant: Q1 (0°–90°): All positive. Q2 (90°–180°): Sine (and csc) positive. Q3 (180°–270°): Tangent (and cot) positive. Q4 (270°–360°): Cosine (and sec) positive. This is because: Q2 has x < 0, y > 0 → cos < 0, sin > 0, tan < 0. Q3 has x < 0, y < 0 → both negative, but tan = y/x > 0. Q4 has x > 0, y < 0 → cos > 0, sin < 0.
• What is a reference angle? ▼
  A reference angle is the acute angle (0° to 90°) between the terminal side of an angle and the nearest x-axis. It is used to find trig values in any quadrant: the magnitude of sin, cos, or tan at angle θ equals the corresponding value at the reference angle; the sign is determined by the ASTC rule. For example, 150° has reference angle 30° (since 180°−150°=30°). So cos(150°) = −cos(30°) = −√3/2 (negative because Q2 cosine is negative) and sin(150°) = +sin(30°) = 1/2 (positive because Q2 sine is positive).
• How do I find the angle from coordinates on the unit circle? ▼
  Given a point (x, y) on the unit circle, the angle θ is found using the two-argument arctangent: θ = atan2(y, x). This function correctly handles all four quadrants unlike the single-argument arctan. Example: point (−√3/2, 1/2) → θ = atan2(0.5, −0.866) = 150°. The result is in (−180°, 180°]; add 360° to convert negative angles to the [0°, 360°) range. Verify: the coordinates for 150° are (cos 150°, sin 150°) = (−√3/2, 1/2). ✓ This calculation also works for off-circle points by normalizing first.

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