Six Names for One Point

Put a point on a circle and push it round. Measure how high it sits, and how far along. That is trigonometry — all of it. Everything else in the subject is bookkeeping about those two lengths.

the angle θ: 60° = π/3 rad
sin θ — the height: √3/2 ≈ 0.866
cos θ — the reach: 1/2 = 0.500
tan θ — on the touching line: √3 ≈ 1.732
sin²θ + cos²θ: 0.750 + 0.250 = 1 — always
arc walked from 0°: 1.047 radii

Drag anywhere on the circle, or grab the wave and crank the walk back and forth. Arrow keys step one degree; shift-arrow jumps corner to corner. Every number on this page is read off the one point — nothing is looked up in a table.

The sixteen corners

The table of “special angles” is not a list of facts. It is sixteen parking spots for the same point — the angles where the lengths come out in closed form. Press one to park there.

θ	radians	sin θ	cos θ	tan θ
0°	0	0	1	0
30°	π/6	1/2	√3/2	√3/3
45°	π/4	√2/2	√2/2	1
60°	π/3	√3/2	1/2	√3
90°	π/2	1	0	—
120°	2π/3	√3/2	−1/2	−√3
135°	3π/4	√2/2	−√2/2	−1
150°	5π/6	1/2	−√3/2	−√3/3
180°	π	0	−1	0
210°	7π/6	−1/2	−√3/2	√3/3
225°	5π/4	−√2/2	−√2/2	1
240°	4π/3	−√3/2	−1/2	√3
270°	3π/2	−1	0	—
300°	5π/3	−√3/2	1/2	−√3
315°	7π/4	−√2/2	√2/2	−1
330°	11π/6	−1/2	√3/2	−√3/3

The three knobs

y = 1.00 · sin(1.00 t + 0)one full cycle every 2π/ω = 6.28

A — the radius1.00a bigger circle throws a taller shadow — amplitudeω — the walking speed×1.00a faster walk packs more cycles into the same time — frequencyφ — the head start0starting further round shifts the whole wave sideways — phase

Most people meet this the other way up: six functions with strange names, a mnemonic about triangles, a table of values to memorize, and — years later, if at all — the remark that it was really about a circle the whole time. The instrument above inverts the order. The circle is the subject. The point’s height is called sine, its reach is called cosine, and the other four names are lengths you can switch on and watch. Nothing on the page is computed any other way.

Two things are worth noticing before reading further. The unrolled waves are not a second fact about sine and cosine; they are the same two lengths with the walk laid out into a ribbon, drawn at the same scale — the wave crests at exactly the circle’s radius, because it is the radius, caught at the top of the turn. And the green arc is the angle: measured not in degrees but in how much rim the point has actually walked, in units of the radius. An angle of one radian is a walk of one radius along the rim. That is the entire secret of radians, and it is why $2\pi$ keeps appearing — it is the circumference.

Where the names come from

The names are older than the functions, and they are names of lines.

Turn on tangent · secant above. The tangent value lives on the line that touches the circle at the point where the walk began — tangere, to touch. The secant is measured along the ray through the moving point, which cuts the circle on its way out — secare, to cut. Park the point at 90° and watch: the ray turns parallel to the touching line, the meeting point runs off past any number you name, and at exactly 90° the two lines simply never meet. “Tan 90° is undefined” is not a rule about a forbidden operation. It is two lines failing to cross.

The co- names are the same constructions built from the complement — the angle’s partner that makes up 90°. The cosine of θ is the sine of 90° − θ, which is why the pair trade values as the point climbs.

Sine has the strangest history of the six: Sanskrit jyā, bowstring, for the chord the angle subtends; transliterated into Arabic as jība; misread centuries later as jayb, meaning fold or bay; and so translated into Latin as sinus. The name is a fossil of a copying error. The length was always just the height of a point on a circle.

The identities that cannot break

Drag the point anywhere. The readout keeps insisting that

\sin^2\theta + \cos^2\theta = 1

and the figure shows why it could not do otherwise: height and reach are the two legs of a right triangle whose hypotenuse is the radius, and the radius is

The most famous identity in trigonometry is the Pythagorean theorem wearing a different shirt.

The same figure gives away the next one for free. With tangent · secant on, look at the triangle formed by the touching line: one leg is the radius to the touching point, length 1; the other leg is the tangent segment; the hypotenuse is the secant. Pythagoras again, on the bigger triangle:

1 + \tan^2\theta = \sec^2\theta

Nothing was derived. Both identities are visible at once, in one picture, at every angle you can drag to.

SOH-CAH-TOA, retired

The triangle definitions are not wrong — they are the circle seen through a keyhole. Inside the first quarter turn, the radius is the hypotenuse of a right triangle with the height and reach as its legs, and “opposite over hypotenuse” is just the height, because the hypotenuse is 1. Scale any right triangle so its hypotenuse is 1 and you have placed it inside this circle.

The keyhole is the problem. Triangles run out at 90°: no right triangle has an obtuse angle in it, let alone a negative one, or a fifth full turn. The point has no such difficulty. It keeps walking, the lengths keep being lengths (now signed, since height below the axis counts negative), and the functions stay defined for every angle that will ever appear in a physics course. The circle is not the advanced view. It is the definition, and the triangle was the special case.

The second panel above is the rest of a precalculus course. Every sinusoid anyone will ever ask you to graph is $y = A\sin(\omega t + \varphi)$ , and the three parameters are not algebra — they are three things you can do to a walking point. Grow the circle and the wave grows with it, because the wave’s height is the radius. Walk faster and the cycles pack tighter. Start further round the rim and the whole wave slides sideways. Amplitude, frequency, phase: three knobs, each a physical act, none of them mysterious once the circle is in view.

Where this goes is worth one sentence. Bolt a second, smaller circle to the rim of the first and let both walk, and the traced curve is the sum of two waves; keep adding circles and you can trace out any repeating curve you could draw with a pen — which is Fourier’s theorem, and the reason every sound you have ever heard, every radio signal, and every image compression scheme is, underneath, circles added to circles. The point walking above is the first term of the series that describes everything that repeats.