Unit Circle Reference Table
| Degrees | Radians | sin θ | cos θ |
|---|---|---|---|
| 0° | 0 | 0 | 1 |
| 30° | π/6 | 1/2 | √3/2 |
| 45° | π/4 | √2/2 | √2/2 |
| 60° | π/3 | √3/2 | 1/2 |
| 90° | π/2 | 1 | 0 |
| 120° | 2π/3 | √3/2 | -1/2 |
| 135° | 3π/4 | √2/2 | -√2/2 |
| 150° | 5π/6 | 1/2 | -√3/2 |
| 180° | π | 0 | -1 |
| 270° | 3π/2 | -1 | 0 |
| 360° | 2π | 0 | 1 |
How it works
The unit circle is the circle of radius 1 centred at the origin of the coordinate plane. It provides the foundational reference for trigonometric function values at all standard angles. Every trigonometric identity, inverse function, and periodic relationship can be derived from the unit circle. The Unit Circle Reference Table displays exact values of sin, cos, and tan for all standard angles in both degrees and radians.
**Key values (memorisation guide)** The pattern for sin at 0°, 30°, 45°, 60°, 90°: sin(θ) = √(n)/2 where n = 0, 1, 2, 3, 4 respectively. So sin(0°)=0, sin(30°)=½, sin(45°)=√2/2, sin(60°)=√3/2, sin(90°)=1.
cos(θ) is the reverse: cos(0°)=1, cos(30°)=√3/2, cos(45°)=√2/2, cos(60°)=½, cos(90°)=0.
tan(θ) = sin(θ)/cos(θ): tan(0°)=0, tan(30°)=1/√3=√3/3, tan(45°)=1, tan(60°)=√3, tan(90°)=undefined.
**Sign rules by quadrant (ASTC / "All Students Take Calculus")** Quadrant I (0–90°): All positive. Quadrant II (90–180°): Sine positive. Quadrant III (180–270°): Tangent positive. Quadrant IV (270–360°): Cosine positive.
**Radian equivalents** 30° = π/6; 45° = π/4; 60° = π/3; 90° = π/2; 180° = π; 270° = 3π/2; 360° = 2π. The radian measure of an angle equals the arc length on the unit circle, which is why radians are natural in calculus.
**Beyond 0–360°** Trigonometric functions are periodic: sin(θ + 2π) = sin(θ). The table extends to any angle by reducing modulo 2π. Negative angles work backwards: sin(−θ) = −sin(θ) (odd function); cos(−θ) = cos(θ) (even function).
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Frequently Asked Questions
- Radians make calculus formulas cleaner: the derivative of sin(x) is cos(x) only when x is in radians. In degrees, d/dx[sin(x°)] = (π/180)cos(x°) — a messy factor appears. Similarly, arc length s = rθ only holds when θ is in radians; in degrees it would be s = rθπ/180. Power series expansions (sin(x) = x − x³/3! + x⁵/5! − ...) require radian input. The radian is the 'natural' unit because it equals the arc length divided by the radius on a unit circle — a dimensionless, geometry-based measure that requires no conversion constants.
- The 'hand trick' for sin: hold your hand with fingers spread. Thumb = 0° (sin=0), index = 30° (sin=½), middle = 45° (sin=√2/2), ring = 60° (sin=√3/2), pinky = 90° (sin=1). Pattern: sin(0°, 30°, 45°, 60°, 90°) = √0/2, √1/2, √2/2, √3/2, √4/2 = 0, ½, √2/2, √3/2, 1. For cos: it's the reverse sequence. Another mnemonic: '0, 1, √2, √3, 2 all divided by 2' for sin, reversed for cos.
- The three primary functions have reciprocals: csc(θ) = 1/sin(θ) (cosecant); sec(θ) = 1/cos(θ) (secant); cot(θ) = 1/tan(θ) = cos(θ)/sin(θ) (cotangent). These appear in calculus integrals (∫sec²(x)dx = tan(x)), physics, and engineering. Common values: csc(30°) = 2; sec(60°) = 2; cot(45°) = 1. The unit circle reference table includes all six functions — sin, cos, tan, csc, sec, cot — at all standard angles.
- tan(θ) = sin(θ)/cos(θ). At 90°, sin(90°) = 1 and cos(90°) = 0. Division by zero is undefined. Geometrically: the tangent function represents the slope of the line from the origin to the point on the unit circle. At 90°, the point is directly above the origin; the line is vertical and has infinite slope — which is not a real number. As θ approaches 90° from below, tan(θ) → +∞; from above (91°), tan(θ) → −∞. This discontinuity (vertical asymptote) occurs at θ = 90°, 270°, and every odd multiple of 90°.