Trigonometry

Unit Circle Chart

How to Read the Unit Circle

Every point on the unit circle corresponds to an angle θ\theta measured counterclockwise from the positive x-axis. The coordinates of that point are (cos⁡θ,  sin⁡θ)(\cos\theta,\;\sin\theta). Because the circle has radius 1, the cosine gives the horizontal distance and the sine gives the vertical distance from the origin.

Unit Circle Diagram

The Unit Circle — All 16 Standard Angles

xyIIIIIIIV1-11-10°30°45°60°90°120°135°150°180°210°225°240°270°300°315°330°(0, 0)

Complete Unit Circle Values

DegreesRadianssin⁡θ\sin\thetacos⁡θ\cos\thetatan⁡θ\tan\thetaPoint (x,y)(x, y)
0°0°00001100(1,  0)(1,\; 0)
30°30°π6\dfrac{\pi}{6}12\dfrac{1}{2}32\dfrac{\sqrt{3}}{2}33\dfrac{\sqrt{3}}{3}(32,  12)\left(\dfrac{\sqrt{3}}{2},\; \dfrac{1}{2}\right)
45°45°π4\dfrac{\pi}{4}22\dfrac{\sqrt{2}}{2}22\dfrac{\sqrt{2}}{2}11(22,  22)\left(\dfrac{\sqrt{2}}{2},\; \dfrac{\sqrt{2}}{2}\right)
60°60°π3\dfrac{\pi}{3}32\dfrac{\sqrt{3}}{2}12\dfrac{1}{2}3\sqrt{3}(12,  32)\left(\dfrac{1}{2},\; \dfrac{\sqrt{3}}{2}\right)
90°90°π2\dfrac{\pi}{2}1100undef.(0,  1)(0,\; 1)
120°120°2π3\dfrac{2\pi}{3}32\dfrac{\sqrt{3}}{2}−12-\dfrac{1}{2}−3-\sqrt{3}(−12,  32)\left(-\dfrac{1}{2},\; \dfrac{\sqrt{3}}{2}\right)
135°135°3π4\dfrac{3\pi}{4}22\dfrac{\sqrt{2}}{2}−22-\dfrac{\sqrt{2}}{2}−1-1(−22,  22)\left(-\dfrac{\sqrt{2}}{2},\; \dfrac{\sqrt{2}}{2}\right)
150°150°5π6\dfrac{5\pi}{6}12\dfrac{1}{2}−32-\dfrac{\sqrt{3}}{2}−33-\dfrac{\sqrt{3}}{3}(−32,  12)\left(-\dfrac{\sqrt{3}}{2},\; \dfrac{1}{2}\right)
180°180°π\pi00−1-100(−1,  0)(-1,\; 0)
210°210°7π6\dfrac{7\pi}{6}−12-\dfrac{1}{2}−32-\dfrac{\sqrt{3}}{2}33\dfrac{\sqrt{3}}{3}(−32,  −12)\left(-\dfrac{\sqrt{3}}{2},\; -\dfrac{1}{2}\right)
225°225°5π4\dfrac{5\pi}{4}−22-\dfrac{\sqrt{2}}{2}−22-\dfrac{\sqrt{2}}{2}11(−22,  −22)\left(-\dfrac{\sqrt{2}}{2},\; -\dfrac{\sqrt{2}}{2}\right)
240°240°4π3\dfrac{4\pi}{3}−32-\dfrac{\sqrt{3}}{2}−12-\dfrac{1}{2}3\sqrt{3}(−12,  −32)\left(-\dfrac{1}{2},\; -\dfrac{\sqrt{3}}{2}\right)
270°270°3π2\dfrac{3\pi}{2}−1-100undef.(0,  −1)(0,\; -1)
300°300°5π3\dfrac{5\pi}{3}−32-\dfrac{\sqrt{3}}{2}12\dfrac{1}{2}−3-\sqrt{3}(12,  −32)\left(\dfrac{1}{2},\; -\dfrac{\sqrt{3}}{2}\right)
315°315°7π4\dfrac{7\pi}{4}−22-\dfrac{\sqrt{2}}{2}22\dfrac{\sqrt{2}}{2}−1-1(22,  −22)\left(\dfrac{\sqrt{2}}{2},\; -\dfrac{\sqrt{2}}{2}\right)
330°330°11π6\dfrac{11\pi}{6}−12-\dfrac{1}{2}32\dfrac{\sqrt{3}}{2}−33-\dfrac{\sqrt{3}}{3}(32,  −12)\left(\dfrac{\sqrt{3}}{2},\; -\dfrac{1}{2}\right)

Quick Patterns

Memorizing the entire unit circle is easier once you notice these patterns:

  • Sine values in Quadrant I follow the sequence 0,  12,  22,  32,  10,\; \dfrac{1}{2},\; \dfrac{\sqrt{2}}{2},\; \dfrac{\sqrt{3}}{2},\; 1 (increasing from 0°0° to 90°90°)
  • Cosine values in Quadrant I follow the reverse sequence 1,  32,  22,  12,  01,\; \dfrac{\sqrt{3}}{2},\; \dfrac{\sqrt{2}}{2},\; \dfrac{1}{2},\; 0 (decreasing from 0°0° to 90°90°)
  • Tangent is undefined at 90°90° and 270°270° because cos⁡θ=0\cos\theta = 0 at those angles
  • ASTC rule — which trig functions are positive in each quadrant: All in Q I, Sine in Q II, Tangent in Q III, Cosine in Q IV
  • Reference angles let you reuse Q I values everywhere: subtract from 180°180° for Q II, subtract 180°180° for Q III, subtract from 360°360° for Q IV

Return to Trigonometry for the full topic list.