The table lists 17 standard angles around the unit circle. Each row gives one angle in two unit systems and the exact values of all six trigonometric functions at that angle.
Column structure:
• Angle (degrees) — angle in degrees from 0° to 360° • Angle (radians) — the same angle written as a multiple of π • sin, cos, tan — the three primary trig functions • csc, sec, cot — the three reciprocal functions
Values are written as exact fractions and surds, never decimals. For example, sin(60°)=23, not 0.866… Cells marked Undefined indicate where the function diverges to infinity — this happens for tan and sec at 90° and 270°, and for csc and cot at 0°, 180°, and 360°.
What Makes These Angles Special
These 17 angles are called special angles because their trig values can be written exactly using small integers and the surds 2 and 3 — no decimal approximations required. Every other angle on the unit circle has irrational trig values that cannot be expressed in such closed form.
The first-quadrant special angles are 0°,30°,45°,60°, and 90° (or 0,6π,4π,3π,2π in radians). The remaining 12 entries are reflections of these across the x-axis, y-axis, and origin, producing the full set covering all four quadrants.
The exact values come from two elementary right triangles. The 45-45-90 isosceles right triangle with legs of length 1 has hypotenuse 2, giving the values at 45°. The 30-60-90 triangle — half of an equilateral triangle, with sides 1,3,2 — gives the values at 30° and 60°. The remaining angles 0°,90°,180°,270°, and 360° come directly from points on the coordinate axes.
Deriving the Values from the Unit Circle
On the unit circle, a point at angle θ measured counterclockwise from the positive x-axis has coordinates (cosθ,sinθ). This single fact connects every entry in the table to a geometric position on the circle.
First-quadrant values come from the two right triangles described above. Other-quadrant values inherit those magnitudes with sign changes determined by the quadrant:
• Quadrant II (90° to 180°): sin>0, cos<0 — sine stays positive, cosine flips sign • Quadrant III (180° to 270°): both sin and cos are negative • Quadrant IV (270° to 360°): sin<0, cos>0 — cosine stays positive, sine flips sign
This is the standard ASTC rule (All Students Take Calculus): All functions positive in Q1, only Sine in Q2, only Tangent in Q3, only Cosine in Q4. The remaining three functions follow from the reciprocal and quotient definitions: csc=1/sin, sec=1/cos, and cot=cos/sin, so each inherits its sign from the function it depends on.
For an interactive exploration of these relationships, see the interactive unit circle visualizer.
When You Will Use This Table
Special-angle values come up constantly in algebra, geometry, calculus, and physics. Common scenarios:
• Solving trig equations by hand — equations like sinx=21 have neat solutions at special angles (x=30° and x=150° from this table) • Verifying calculator answers — when a calculator gives 0.7071… for sin(45°), you can confirm it matches the exact value 22 • Definite integrals over standard intervals — integrals from 0 to 2π or 0 to 2π frequently evaluate at these specific angles • Geometry and physics — vector components at 30°,45°, or 60° require exact values to avoid accumulated rounding error • Building unit-circle fluency — memorizing these 17 entries unlocks the entire unit circle and is foundational for trig identities
The biggest practical benefit is staying in exact form. A problem that uses 23 throughout stays clean; the same problem with 0.8660254… accumulates roundoff and obscures algebraic structure.
For computations at non-special angles, use the trigonometry calculator.
Related Tables and Tools
This table is one of several trigonometry references on the site. Pair it with the following for full coverage:
Related tables:
• Trigonometric identities table — 115 identities across 15 families, including Pythagorean, sum-angle, double-angle, half-angle, and reduction formulas • Inverse trigonometric functions table — exact values of arcsin, arccos, and arctan at standard inputs • Sum, difference, double, triple, and half-angle formulas — one focused table per multi-angle family • Negative, complement, and supplement angle formulas — sign-flip and reflection identities • Trigonometric reduction formulas — reduce any angle to a first-quadrant equivalent
Companion tools:
• Interactive unit circle — drag a point around the circle and watch sin, cos, and tan values update in real time • Trigonometry calculator — compute trig values at any angle, special or not • Angle converter — convert between degrees and radians
For the underlying theory, see the trigonometry overview.