Sine and cosine are not table entries to memorise; they are the shadows of a point walking around a circle. Cosine is how far east it has gone, sine how far north, and the whole subject — waves, phases, the angle-sum identities, even the strange multi-valuedness of arcsin — falls out of watching that one point move. The four tabs below share a single angle dial θ, so the point you set on the circle is the same point riding the wave and driving the identity. Everything is computed live from the real geometry; nothing is a canned plot.
Fix a circle of radius 1 centred at the origin and start a ray at the positive x-axis. Rotate the ray anticlockwise by an angle θ. It pierces the circle at a single point P. By definition, the x-coordinate of P is cos θ and the y-coordinate is sin θ. That is the whole of trigonometry's foundation — two names for two coordinates.
Because P lies on the circle of radius 1, its coordinates satisfy x² + y² = 1, which is exactly the identity cos²θ + sin²θ = 1. It isn't a separate fact to memorise; it is the equation of the circle read aloud.
The tangent is the slope of the ray from the origin to P, rise over run, sin θ / cos θ. Geometrically it equals the height at which the ray crosses the vertical line x = 1 — the segment drawn in purple. As θ approaches 90° the run (cos θ) collapses to zero, the ray becomes vertical, and tan θ races to infinity.
The reference angle is the acute angle between the ray and the x-axis. In every quadrant the sizes of sin, cos and tan are the reference angle's values; only the signs change, following the sign of x (cos) and y (sin) in that quadrant. This is why "all-students-take-calculus" mnemonics work — they are just tracking which coordinate is negative.
Measuring θ in radians makes the circle's own arc length the unit: an angle of θ radians subtends an arc of length θ on the unit circle. A full turn is 2π because the circumference is 2π. Radians are what make calculus of sine clean — the derivative of sin is cos only when angle is measured this way.
Finally, rotating past 360° returns P to a previous position, so sin and cos repeat with period 2π. Periodicity is geometric: it is the statement that going all the way around brings you home.
Let a point move around a circle of radius A at a steady angular rate k, starting from an angle offset φ. At "time" x its angle is kx + φ, so its height is A·sin(kx + φ). Plotting that height against x is precisely the wave on the right. The three sliders map one-to-one onto three physical facts about the circle: how big it is, how fast it turns, and where it started.
Amplitude A is the radius, the maximum height reached. Angular frequency k is radians of turn per unit x; the wave completes one full cycle when kx advances by 2π, giving a period of 2π/k and an ordinary frequency of k/2π cycles per unit. Phase φ shifts the pattern along x by −φ/k without changing its shape.
The cosine partner is the same motion read on the other axis. Since the x-shadow always leads the y-shadow by a quarter turn, cos θ = sin(θ + 90°): cosine is sine with a 90° head start. That single phase relationship is why sines and cosines can always be traded for one another.
This "circle unrolled" picture is the doorway to Fourier analysis: any repeating signal, however jagged, is a sum of such circular motions at different radii, rates and head starts. Sound, light, tides and alternating current are all read this way — each is a chorus of rotating points, and the wave you see is their combined shadow.
Start with a right triangle whose hypotenuse is the unit ray at angle a+b. We build it in two stages. First rotate by b and draw a small right triangle with hypotenuse 1; its far corner sits at height sin b and horizontal reach cos b. Then, on top of that triangle, we rotate by a further a.
The clever move is to erect a second right triangle on the slanted top edge of the first. Projecting the whole staircase onto the vertical axis, the total height splits cleanly into sin a·cos b (the rise contributed by the long horizontal leg of length cos b, tilted by a) plus cos a·sin b (the rise contributed by the short vertical leg of length sin b). Their sum is the height of the outer ray, which is sin(a+b).
Projecting the same staircase onto the horizontal axis gives the cosine identity. Here the two contributions point in opposite directions — cos a·cos b reaches forward while sin a·sin b pulls back — so they subtract: cos(a+b) = cos a cos b − sin a sin b. The lone minus sign that trips up students is simply two legs pointing opposite ways.
Every other identity is a corollary of these two. Setting b = a gives the double-angle formulas. Adding and subtracting the sine and cosine versions gives the product-to-sum rules that turn multiplication of waves into interference. And the same construction, done with the complex exponential ei(a+b) = eiaeib, produces both identities in one line — the diagram and the algebra are the same statement.
A function can be inverted only if it is one-to-one — each output must come from a single input. Sine is emphatically not: it is periodic, so any height it reaches is reached again every full turn, and within a single turn it reaches most heights twice (once climbing, once falling). Feeding a height back through "which angle?" therefore returns an infinite family.
To manufacture a genuine inverse we amputate the domain, keeping only the stretch [−90°, 90°] where sine rises steadily from −1 to 1 without repeating. On that restricted piece sine is one-to-one, and its inverse — arcsin, with outputs confined to that same interval — is well defined. This chosen output range is called the principal branch.
The full solution set is easy to reconstruct from the principal value α = arcsin y. Because sine is symmetric about 90°, the other solution in a turn is 180° − α, and because it is periodic, you may add any whole number of full turns. So x = α + 360°k or x = (180° − α) + 360°k for any integer k — exactly the crossings you see on screen.
The merging of solutions as y → 1 is the geometric reason arcsin has infinite slope at the endpoints: near a crest the sine is nearly flat, so a tiny change in height corresponds to a large change in angle. The same amputation strategy defines arccos (branch [0°, 180°]) and arctan (branch (−90°, 90°)); every inverse trig function is a deliberate choice of one slice from a repeating pattern.