Enter an angle to find exact sine, cosine, and tangent values with step-by-step derivation
Angle (degrees)
Result
Angle:
sin
-
cos
-
tan
-
Detailed Derivation
Exact Trig Values for Common Angles
0: sin=0, cos=1, tan=0
30: sin=1/2, cos=sqrt(3)/2, tan=1/sqrt(3)
45: sin=sqrt(2)/2, cos=sqrt(2)/2, tan=1
60: sin=sqrt(3)/2, cos=1/2, tan=sqrt(3)
90: sin=1, cos=0, tan=undefined
Exact trigonometric values for special angles are derived from 30-60-90 and 45-45-90 triangles. These values are exact expressions, not decimal approximations, making them essential for precise mathematical work.
⚠For non-special angles, approximate decimal values are shown. Exact forms are available for multiples of 15 (15, 30, 45, 60, 75, etc.).
What Are Exact Trig Values?
Exact trig values are precise expressions that give the sine, cosine, and tangent of special angles without rounding. They come from the geometry of special right triangles and the unit circle.
30-60-90 Triangle
Sides: 1, sqrt(3), 2. Opposite 30=1, opposite 60=sqrt(3). Ratios give standard exact values.
45-45-90 Triangle
Sides: 1, 1, sqrt(2). Both acute angles are 45. sin45 = cos45 = 1/sqrt(2) = sqrt(2)/2.
Reference Angles
Any angle in QII-QIV uses its acute reference angle plus the appropriate sign (+/-) based on quadrant.
Unit Circle
Coordinates (cos theta, sin theta) on the unit circle. Quadrantal angles (0,90,180,270) have coordinates (0,+/-1) or (+/-1,0).
Free online calculators and tools covering mathematics, unit conversion, text processing, and daily life. Accurate, fast, mobile-friendly, and completely free to use.