Unit circle explorer

The unit circle is the circle of radius 1 centred on the origin. Rotate a point around it by an angle θ from the positive x-axis, and its coordinates are (cos θ, sin θ). That is the definition — everything else follows.

Why sin² + cos² = 1

The radius is 1, and the point sits at (cos θ, sin θ), so Pythagoras on the right triangle formed by the two legs gives cos²θ + sin²θ = 1² immediately. The most fundamental trigonometric identity is just the theorem you already knew, applied to this diagram.

Where the waves come from

Trace the point around the circle and plot its height against the angle: that curve is the sine wave. Plot its horizontal position and you get cosine. They are the same shape shifted by 90°, because moving a quarter turn around the circle swaps the roles of the two legs.

Why tangent has asymptotes

tan θ = sin θ / cos θ. At 90° and 270° the cosine is zero, so the tangent is undefined and the curve shoots to infinity. Geometrically, tan θ is the length of the tangent segment from the point to the x-axis — which becomes infinite when the radius is vertical.

Radians are the natural unit

One radian is the angle that cuts an arc equal in length to the radius, so a full circle is 2π radians. Degrees are a Babylonian convention with no mathematical significance. Radians are what makes d/dx sin x = cos x true; in degrees that derivative carries an ugly π/180 factor.

The values worth memorising

Only three: 30°, 45° and 60°, with values 1/2, √2/2 and √3/2. Everything else is one of those with a sign determined by the quadrant. The mnemonic “All Students Take Calculus” gives which functions are positive in quadrants I, II, III and IV — all, sine, tangent, cosine.