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Interactive Unit Circle

Drag the blue point around the circle and see sin, cos and tan change as you move. Each value is drawn as a colored line segment, so you can see where it comes from instead of memorizing a chart. Everything runs in your browser.

How to use the unit circle

Drag the blue point around the circle with a mouse or your finger. It snaps to whole degrees, so 90° and 270° are reachable exactly. The readout at the top (θ 45° at the start) follows the point.
Or type an angle into the Angle box. It accepts 0 to 360 in degrees and 0 to 6.2832 in radians, and a hint appears if you go outside that range.
For fine steps, focus the point and press the arrow keys to move 1 degree, Page Up or Page Down to move 15 degrees, or Home to go back to 0.
Read the values panel: the sin θ, cos θ and tan θ cards update at every position, to 4 decimals.

What the colored segments mean

The circle has radius 1, so the point at angle θ, measured counter-clockwise from the positive x-axis (the small arc), is at (cos θ, sin θ). Each function matches one segment and one card.

sin θ (orange): the vertical segment from the x-axis up to the point. Its length is the point's height (y).
cos θ (green): the horizontal segment along the x-axis from the center. Its length is the point's width (x).
tan θ (purple): the vertical segment on the line x = 1, the tangent line. A dashed ray from the center through the point reaches it, and tan θ is the height where the ray lands, which is why tan θ = sin ÷ cos.

A segment on the wrong side of an axis means a negative value: the cos segment points left in the second and third quadrants, and the sin segment points down in the third and fourth.

Unit circle sin, cos and tan values at common angles

These are the numbers the cards show at some familiar angles. Drag to each one to check.

Angle	sin θ	cos θ	tan θ
0°	0.0000	1.0000	0.0000
30°	0.5000	0.8660	0.5774
45°	0.7071	0.7071	1.0000
60°	0.8660	0.5000	1.7321
90°	1.0000	0.0000	Undefined
120°	0.8660	-0.5000	-1.7321
210°	-0.5000	-0.8660	0.5774

Why tan is Undefined at 90° and 270°

At 90° and 270° the point is directly above or below the center, so cos θ is 0 and the division sin ÷ cos has no answer. The dashed ray never reaches the line x = 1, so the purple segment disappears and the card reads Undefined. The same happens to sec θ at those two angles. With csc θ and cot θ, it is sin θ that is 0, at 0°, 180° and 360°.

Degrees and radians

Use the Unit switch to change the readout and the Angle box between Degrees and Radians. Radians show as decimals: 90° is 1.5708 rad, 180° is 3.1416 rad and a full turn is 6.2832 rad. The drawing and the values do not change, only how the angle is written.

Secant, cosecant and cotangent

Tick Show secant, cosecant and cotangent to add the three reciprocal functions: sec θ (1 ÷ cos), csc θ (1 ÷ sin) and cot θ (cos ÷ sin). On the drawing, sec is a crimson segment on the x-axis, csc a teal segment on the y-axis and cot an amber segment along the line y = 1, together with a dashed tangent to the circle at the point.

Frequently asked questions

How does tangent work on the unit circle?

Tangent is sin ÷ cos, and on the circle it is a length you can see. Draw a ray from the center through the point and extend it to the vertical line x = 1: the height where it lands is tan θ. The purple segment on the drawing is exactly that. At 45° it lands at height 1, so tan 45° is 1.0000.

Why is tan 90 undefined?

At 90° the point sits straight above the center, so cos θ is 0 and tan θ = sin ÷ cos would divide 1 by 0. Geometrically, the ray through the center and the point runs parallel to the line x = 1 and never meets it, so there is no tan segment to draw. Drag to 90° or 270° and tan θ reads Undefined while the purple segment disappears.

Why is cos 90 zero?

cos θ is the horizontal position (x) of the point. At 90° the point is directly above the center, so its x position is 0 and the green cos segment shrinks to nothing. The card shows 0.0000, and sin θ is 1.0000 because the point is at full height.

What are secant, cosecant and cotangent?

They are the reciprocals of the main three: sec θ = 1 ÷ cos θ, csc θ = 1 ÷ sin θ and cot θ = cos ÷ sin. Turn on Show secant, cosecant and cotangent to add their cards and their segments to the drawing. sec is Undefined at 90° and 270°, while csc and cot are Undefined at 0°, 180° and 360°.

How do I use the unit circle to find sin and cos?

Read them off the point's coordinates. The point on the circle at angle θ is (cos θ, sin θ), so cos θ is how far right or left of the center it sits and sin θ is how far above or below the x-axis. At 30° the cards show sin 0.5000 and cos 0.8660. Left of the center and below the axis the values turn negative, as at 210°, where both are negative.

Can I use radians instead of degrees?

Yes. Switch the Unit control from Degrees to Radians and the θ readout and the Angle box use radians, from 0 to 6.2832. Radians appear as decimals rather than multiples of π, so 90° reads 1.5708 rad and 180° reads 3.1416 rad.

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