Coordinates are (cos, sin)
The horizontal coordinate is cosine and the vertical coordinate is sine. That single fact drives the whole circle.
EXPLORE · CONNECT · RECALL
Unit Circle
Drag an angle, read exact trig values, connect the same point to sine and cosine waves, then train the 16 standard angles.
16exact landmarks
6trig functions
localpractice history
INTERACTIVE EXPLORER
See what the angle means
Click or drag the circle. Keyboard: ←/→ 1°, Shift + ←/→ 15°.
Point direction
Enter any non-zero (x, y); it is normalized onto radius 1.
EXACT STANDARD ANGLE45° · π/4
Quadrant I · reference angle 45°
Point (cos θ, sin θ)(√2/2, √2/2)(0.707107, 0.707107)
CIRCLE → WAVES
height = sin θ · horizontal = cos θThe same angle on sine and cosine
sin θcos θθ = 45°
16-ANGLE MASTER CHART
Exact values in one turn
| Degrees | Radians | (cos, sin) | sin | cos | tan |
|---|
Click a row to load that angle. Hide answers to turn the chart into a recall sheet.
DAILY 12
Recall the circle from different directions
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RADIANS
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FOCUSED PRACTICE
Pick the skill that is slowing you down
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MIXED
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30sLearn 30°, 45°, 60° once
Use the first-quadrant magnitudes 1/2, √2/2 and √3/2, then reflect them around the circle.
Quadrants decide the signs
I: all positive. II: sin positive. III: tan positive. IV: cos positive.
Reference angles keep it small
Reduce any standard angle to 30°, 45° or 60°, then apply the quadrant sign.
Exactness and scopesymbolic standard angles · decimal arbitrary angles
The 16 standard landmarks use a fixed verified table of exact fractions and radicals. Arbitrary angles use browser trigonometric functions and are shown as decimal approximations, never relabeled as exact radicals.
Tangent and secant are undefined when cosine is zero; cosecant and cotangent are undefined when sine is zero. Axis cases are handled directly so the page does not display giant floating-point artifacts near 90°, 180°, or 270°.
Why the unit circle creates sine and cosine wavessame point · different view
As θ turns around a radius-1 circle, the point is (cos θ, sin θ). Track its vertical coordinate through time and you get the sine wave; track its horizontal coordinate and you get the cosine wave.