Interactive unit circle: drag the point and read sin, cos, and tan.
Interactive Unit Circle simulator
Drag the point around the circle
This simulation needs a modern browser with canvas support. The measurements and formula below still describe the physics.
Live measurements
Controls
Graph
Formula
(x, y) = (cos θ, sin θ)
- θ Angle
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- x cos θ
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- y sin θ
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Observations
What's happening
Real-world examples
The 45° triangle
Drag to 45° and the two legs become equal: sin θ and cos θ are both √2/2, about 0.707, and tan θ = 1.
Crossing 90°
Approach 90° and watch cos θ shrink to 0 while tan θ explodes: division by a vanishing cosine has no answer.
Quadrant signs
Tour the four quadrants and watch the signs flip: sin θ is positive in the top half, cos θ in the right half.
The sine wave
Set a rotation speed and the point traces a perfect sine wave on the right: circular motion seen edge-on.
▸ More about
Uses of unit circle
- Learning or teaching trigonometry with a live picture instead of a static chart
- Memorizing the special angles before an exam by dragging to each one
- Understanding rotations when programming games or graphics
- Seeing where the sine wave in physics and audio actually comes from
unit circle pitfalls
- Forgetting that cos θ turns negative in quadrants II and III
- Mixing radians and degrees in the same calculation
- Reading tan θ as a coordinate of the point instead of a ratio
unit circle questions (9)
What is an interactive unit circle calculator?
An interactive unit circle calculator lets you drag a point around a circle of radius 1 and read sin, cos, and tan as live values. The horizontal coordinate is cos θ and the vertical is sin θ. Open the explorer, drag to any angle, and watch the reference triangle and the readouts update together.
What is the unit circle?
The unit circle is a circle of radius 1 centered at the origin. Its power is that the point at angle θ sits at coordinates (cos θ, sin θ), so one picture defines sine and cosine for every angle, not just the acute angles of a right triangle. The explorer draws the angle, the reference triangle, and the live values as you drag, so the definition becomes something you can feel rather than memorize.
How do I read sin cos tan on the unit circle?
Read the coordinates of the point: the horizontal coordinate is cos θ and the vertical coordinate is sin θ. For example, drag the point to 30° and the readouts show cos θ = 0.866 and sin θ = 0.5, which are the exact values √3/2 and 1/2. Tangent is not a coordinate: it is the ratio sin θ / cos θ, shown as its own live measurement. The reference triangle drawn inside the circle makes the two legs visible.
Why is tan undefined at 90 degrees?
Because tan θ = sin θ / cos θ, and at 90° the cosine is exactly 0, so the division has no value. Drag the point toward 90° and watch tan θ grow without bound while cos θ shrinks: that runaway growth is the vertical asymptote of the tangent function. The same thing happens at 270°, the other angle where the point crosses the y-axis.
What is the difference between radians and degrees?
Degrees split a full turn into 360 steps by convention; radians measure the same turn by arc length, so a full circle is 2π radians because the circumference of a unit circle is 2π. The explorer shows both readouts at once. Radians look odd at first but make the math natural: a 1 radian angle cuts an arc of length exactly 1 on the unit circle, and calculus formulas for sine and cosine only take their clean form in radians.
What are the special angles I should memorize?
The classic set is 30°, 45°, and 60°, whose sine and cosine values are 1/2, √2/2, and √3/2 in different orders, plus the axis angles 0°, 90°, 180°, and 270°. Every other named angle on the standard unit circle chart is one of these reflected into another quadrant using a reference angle. Use the presets to jump straight to each special angle and watch which leg of the triangle carries which value.
What is a reference angle and how do I find it?
The reference angle is the acute angle between the radius arm and the x-axis, always between 0° and 90°. To find it: subtract from 180° in quadrant II, subtract 180° in quadrant III, and subtract from 360° in quadrant IV. For example, 150° has reference angle 30°, so its sine and cosine have the same magnitudes as at 30° with only the signs changed. Drag through the quadrants and watch the triangle repeat itself in mirror image.
How do I convert radians to degrees?
Multiply by 180/π. So π/3 radians × 180/π = 60°, and going the other way, degrees convert to radians by multiplying by π/180. The conversion works because a full turn is both 360° and 2π radians, so the two units are locked in that ratio. The explorer shows both readouts side by side, which makes the common conversions (π/6 = 30°, π/4 = 45°, π/2 = 90°) automatic with repetition.
Why can sin and cos never be greater than 1?
Because they are coordinates of a point stuck on a circle of radius 1. The point can never be further than 1 unit from either axis, so both coordinates live in the range -1 to 1 forever. Tangent has no such cage: it is the ratio sin θ / cos θ, and as cos θ shrinks near 90° the ratio grows without limit. If a calculation hands you sin θ = 1.4, something upstream is wrong.