Unit circle
Interactive unit circle: drag the angle and see sin, cos and tan at once. The best way to understand trigonometry.
Drag the point along the circle, or enter the angle below.
What the unit circle is
The unit circle is a circle of radius 1 centred at the origin of the coordinate system. It's the clearest way to understand trigonometric functions. Moving along the circle by an angle α (measured from the positive horizontal axis, counter-clockwise), we get a point whose coordinates are exactly (cos α, sin α).
How to read the functions off the circle
- xCosine is the horizontal (x) coordinate of the point – how far left/right it lies.
- ySine is the vertical (y) coordinate of the point – how high it lies.
- /Tangent is the ratio sin/cos – the slope of the line from the origin to the point.
Why sin²α + cos²α = 1
A point on the circle is exactly 1 unit from the centre (that's the radius). By the Pythagorean theorem, the square of this distance equals the sum of the squares of the coordinates: cos²α + sin²α = 1². This relationship is the foundation of all trigonometry, and you can see it at a glance on the circle.
Signs in the quadrants
The circle nicely shows when the functions are positive and negative. In the first quadrant (0–90°) everything is positive. In the second (90–180°) only sine is positive. In the third (180–270°) tangent is positive. In the fourth (270–360°) only cosine. Mnemonic: “All – Sine – Tangent – Cosine”.
Tip: try entering an angle greater than 360° or negative – the circle keeps spinning because the functions are periodic.