Sine: carrying the opposite across
On the unit circle the hypotenuse is 1, so sinθ = opposite ÷ hypotenuse = opposite. The vertical red side is the sine. Tap Step +15° to carry that height straight across onto the graph. The graph runs from −360° to 720° — scroll sideways to watch it repeat.
Why is the opposite equal to sinθ?
In any right-angled triangle SOH tells us sinθ = opposite / hypotenuse. On the unit circle the radius (the hypotenuse) is set to 1, so dividing by 1 changes nothing — the opposite side and sinθ are the same length.
As the radius sweeps round, the height of the point above the horizontal axis is exactly sinθ. Plotting that height against the angle traces the sine wave. Past 360° the radius is back where it started, which is why the graph repeats every 360°.
Cosine: standing the adjacent up
Cosine is the horizontal side: cosθ = adjacent ÷ hypotenuse = adjacent. To read it on the same vertical scale as the graph we swing it up through a quarter turn (the arc), then carry it across. Notice the cosine graph is just the sine graph shifted — cosθ = sin(θ + 90°).
Why swing the adjacent up?
The adjacent side lies along the horizontal axis, but a graph plots heights up the page. The quarter-turn arc rotates that horizontal length onto the vertical axis without changing it — now it can be carried straight across, just like the sine.
Because the adjacent side leads the opposite side by a quarter turn, the whole cosine curve sits 90° ahead of the sine curve.
Tangent: the line that touches the circle
Draw the vertical line that just touches the circle at (1, 0) — the tangent line. Extend the radius until it hits that line; the radius extended is the secant and the piece of the tangent line it cuts off is tanθ. From similar triangles tanθ = sinθ ÷ cosθ. As θ nears 90° the radius becomes parallel to the line, so tanθ shoots off to infinity — that is the asymptote.
Where does tanθ = sinθ / cosθ come from?
The little triangle inside the circle has sides cosθ (across), sinθ (up) and 1 (radius). The big triangle out to the tangent line has base 1 (a full radius), height tanθ and slant secθ. The two triangles share the angle θ, so they are similar.
Matching height-to-base ratios: sinθ / cosθ = tanθ / 1, which gives tanθ = sinθ / cosθ. Matching slant-to-base ratios gives secθ = 1 / cosθ.
The reciprocal functions on one circle
Every length here lives on the unit circle. tanθ and secθ sit on the vertical tangent line; cotθ and cosecθ sit on the horizontal tangent line at the top. The tangent and cotangent lines together box in a rectangle. Drag the angle, then read the three nested similar triangles underneath.
The three triangles below are all similar (each contains the angle θ). Read across them and the reciprocal relationships drop out for free.
The three similar triangles
Start with the inner triangle. Divide every side by cosθ to get the tangent triangle; divide every side by sinθ to get the cotangent triangle.
Sketching a reciprocal graph
To sketch cosecθ you only need its parent. Reveal it one idea at a time: where the parent is zero the reciprocal has an asymptote; where the parent reaches ±1 the two curves touch; everywhere else the reciprocal is the “flip” of the parent. Scroll to see it repeat.
Proofs & identities
The same picture proves the identities. Switch between the basic functions and the reciprocal functions; drag the angle and watch both sides of each identity stay equal.