Unit circle and waves
Drag round the unit circle and watch it unwrap into sin and cos, with quadrants, reference angles and exact values; find every solution of sin x = k; see radians as arc length and sin θ/θ → 1; fit A sin(B(x − C)) + D to the book's daylight and tide data; watch simple harmonic motion as the shadow of circular motion; and aim a turret with atan, atan2, y-down screens and the degrees bug.
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How to use it
Trigonometry for every angle comes from one picture: a point P going round a circle of radius 1.
Its x-coordinate is cos θ and its y-coordinate is sin θ. Each tab shows one consequence of that;
angles can be typed in degrees or radians (150, 5pi/6, -30, 11pi/6) — choose the unit
with the degrees / radians switch.
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Unit circle — the “Spin the unit circle” lab: drag P (it snaps to 15°; untick to move freely) or press Spin. The quadrant is shaded, the reference angle α is drawn, and beside the circle sin θ and cos θ unwrap into waves as θ grows — past 360° and below 0° too. Pick sin, cos or tan to see the book’s reference-angle procedure step by step, with exact values (½, √2/2, √3/3, …) at the special angles. With tan, the tangent line x = 1 shows where the name comes from.
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Solve sin x = k — every solution of sin x = k, cos x = k or tan x = k: on the circle (the two angles in one turn) and on the graph from −2π to 4π. The principal value a calculator returns is circled; the other solution comes from symmetry, and the general solution is written out.
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Radians — an angle as arc length: the arc s = rθ, the sector area ½r²θ, one radian marked on the circle, conversions both ways, and the small-angle limit: sin θ/θ → 1 in radians but π/180 in degrees.
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Sinusoids — build y = A sin(B(x − C)) + D (or with cos) and fit it to the book’s data with its procedure: daylight hours at 40° N, a harbour tide, or UK mains. A threshold shades where the curve is above a level — the tide window for a boat that needs 2.0 m of water.
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Harmonic motion — a mass on a spring as the shadow of a point going round a circle: ω = √(k/m), the period and frequency, x(t) traced against time, and the pendulum’s T = 2π√(L/g).
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atan2 turret — the “Rotate and aim” lab: a turret aims at a target you drag. Switch between
atan(dy/dx)(which points the wrong way whenever dx < 0) andatan2(dy, dx); tick y points down to see screen coordinates turn positive angles clockwise; tick degrees bug to pass degrees to cos and sin. The code the turret runs is shown line by line.
Open an example from the book (above the tabs) loads any of the book’s worked examples.
⛶ Full screen (or F) gives the current tab’s circle or graph the whole screen — the browser’s
full screen where it is allowed, otherwise a full-window layer (iPhone); a graph beside it grows to
match, and everything is redrawn as the window changes. Escape leaves.
Limits
Values are double-precision decimals; exact values are shown for multiples of 30° and 45°. The sinusoid fit is the book’s procedure from a maximum, a minimum, the period and when the maximum occurs — not a least-squares fit to many points. The harmonic-motion animation runs in slow motion; the numbers are the real ones.
Related: Complex numbers, visually (Euler’s formula, where cos and sin become one exponential), Graphing calculator (plot any trig function, or polar curves r = f(θ)), Geometry (solve triangles), Fourier series (sums of sines), Algebra step by step (its Transformations tab slides a, b, h and k on sin x) and Physics.
Companion to the chapter Trigonometry of Mathematics, Reconnected.
Linking to an example
Share link copies a URL that reopens the tool as you have it. Links can also be written by hand
with these parameters, followed by #tab to choose the tab (#circle, #solve, #radians,
#waves, #shm, #aim). Settings a link leaves out take their defaults. A + in a link means
plus (%2B works too); write a space as %20, or leave spaces out.
| Parameter | Meaning | Example |
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the angle θ, in the unit given by |
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the right-hand side of fn(x) = k |
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radius for arc length and sector area |
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A, B, C, D of the sinusoid; expressions allowed |
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the data: maximum, minimum, period, and where the maximum is |
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right-hand end of the graph |
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a threshold to shade above (empty for none) |
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mass (kg), spring constant (N/m), amplitude (m), phase (rad) |
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pendulum length (m) |
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the target, relative to the turret |
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a preset from the list below (opens its tab if the link has no |
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For example ?t=150&fn=sin#circle,
?fn=sin&k=1/2&unit=rad#solve and
?aim=atan&tx=-2&ty=-1#aim.
Presets (?example=…):
| Name | Shows |
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the “Spin the unit circle” lab, at 140° (quadrant II) |
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sin 150° = +sin 30° = 1/2 |
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cos 240° = −cos 60° = −1/2 |
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tan(11π/6) = −tan 30° = −√3/3 |
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sin 765° = sin 45° = √2/2 (two full turns) |
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sin x = 1/2: x = π/6 and 5π/6, plus 2πn |
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cos x = −0.222: the inverse gives one answer, symmetry the other |
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one radian: the arc as long as the radius |
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135° = 3π/4 ≈ 2.356 rad |
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a pizza slice: r = 18 cm, θ = π/4, crust 14.14 cm, area 127.2 cm² |
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sin θ ≈ θ for small θ: sin θ/θ → 1 (in radians only) |
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y = 2 sin(2(x − π/4)): amplitude 2, period π, shift π/4 |
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daylight at 40° N: H = 3 sin(2π/365 (t − 80.75)) + 12 |
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the harbour tide and the 2.0 m window over the bar (23:27 to 06:33) |
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UK mains: 325 sin(100πt), period 20 ms |
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0.5 kg on a 200 N/m spring, pulled 4 cm: ω = 20 rad/s, T = 0.314 s |
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the “Rotate and aim” lab: atan(dy/dx) points the turret the wrong way at (−2, −1) |
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atan2(dy, dx) gets (−2, −1) right: −153.43° |
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screen coordinates: y points down, so positive angles turn clockwise |
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the degrees bug: degrees passed to cos and sin |
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the argument of −1 − i: arctan(b/a) says 45°, the truth is −135° |
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