Unit Circle Calculator: Exact sin, cos and tan

How it works

(x, y) = (cos θ, sin θ) · tan θ = sin θ / cos θ · csc = 1/sin, sec = 1/cos, cot = cos/sin · reference angle = acute angle to the x-axis

On the unit circle (radius 1, centred at the origin) the point at angle θ, measured counter-clockwise from the positive x-axis, has coordinates (cos θ, sin θ). Every angle reduces to a coterminal angle between 0° and 360°, and its trig values equal those of the reference angle — the acute angle to the x-axis — with the sign set by the quadrant (All Students Take Calculus: all positive in I, sine in II, tangent in III, cosine in IV). Exact radical values are shown for every multiple of 15°, including the special angles 30°, 45° and 60°.

Worked example

For θ = 150°: it lies in quadrant II, the reference angle is 180° − 150° = 30° (π/6), and the angle in radians is 5π/6. sin 30° = 1/2 and cos 30° = √3/2; in quadrant II sine stays positive and cosine turns negative, so the point is (−√3/2, 1/2). Then tan 150° = −√3/3 ≈ −0.57735, csc = 2, sec = −2√3/3 and cot = −√3.

Sources: Unit circle — Wikipedia · Unit circle — Math is Fun

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