Trigonometric functions

The formula (the core equation(s), each with a one-line reading of what it says)

sin(θ) = opposite / hypotenuse
cos(θ) = adjacent / hypotenuse
tan(θ) = opposite / adjacent = sin(θ) / cos(θ)

Reads: in a right triangle, each function is a ratio of two sides, so it depends only on the angle — a 3-4-5 triangle has the same 36.87° corner whether the sides are inches or kilometers.

Unit circle:  cos(θ) = x,  sin(θ) = y   for the point at angle θ on a circle of radius 1

Reads: the definition that survives past 90° — sweep the angle all the way around and cosine and sine are just the point's coordinates, negative signs and all.

sin²(θ) + cos²(θ) = 1

Reads: Pythagoras on the unit circle; the identity that lets you get one function from the other and the sanity check your code should never fail.

sin(θ + 2π) = sin(θ),   cos(θ + 2π) = cos(θ)

Reads: both repeat every full turn — the property that makes them the alphabet of anything periodic.

d/dθ sin(θ) = cos(θ),   d/dθ cos(θ) = −sin(θ)      (θ in radians)
sin(θ) = (e^(iθ) − e^(−iθ)) / (2i)                  (DLMF 4.14 [5])

Reads: each is the other's rate of change, which is why the solution of x'' = −ω²·x — every undamped oscillator — is a sine; the exponential form is what phasor math runs on.

Where you meet it (2-4 concrete engineering situations, specific: bench, test stand, review board)

Resolving a load on the test stand. A 500 lb pull applied through a cable rigged 35° above horizontal loads the fixture with 500·cos(35°) = 409.6 lb sideways and 500·sin(35°) = 286.8 lb up. Every load cell alignment check, every bolt-pattern shear calc, every free-body diagram in the stress report is cosine for the along component and sine for the across component. Get the angle reference wrong — measured from the wrong axis — and the two swap, which is the single most common markup in a loads review.

Reading AC off the bench. Wall power is v(t) = 169.7·sin(2π·60·t) volts — 120 V is the RMS value, and the peak is 120·√2 = 169.7 V. The √2 exists because the RMS of a sine is 1/√2 ≈ 0.7071 of its peak. Spec an input capacitor for 120 V because that's what the meter said and you're 50 volts short.

Phase between two channels. Two accelerometers on a vibration fixture, both showing 1 kHz, one lagging the other by 30°. That 30° is 30/360 of a period — 83.3 µs — and whether it means a mode shape or a cabling error is a question the review board will ask. Phase angle is nothing but the argument inside sin(ωt + φ), and every Bode plot, every power-factor number, and every beamforming weight lives there too.

The small-angle shortcut. Aero stability derivatives, pendulum periods, beam deflection slopes — anywhere the analysis says sin(θ) ≈ θ and tan(θ) ≈ θ, someone decided the angle stays small. In radians, the error in sin(θ) ≈ θ is 0.13% at 5°, 0.51% at 10°, and 2.1% at 20°. Know which line of that table your model lives on before you defend it.

How it works (the real substance — behavior, gotchas, limits of validity, the mistake people make)

There are two definitions and they hand off to each other. The right-triangle version only works from 0° to 90°, because a right triangle can't hold an obtuse corner next to its right angle. The unit-circle version takes over from there: park a point on a radius-1 circle at angle θ from the positive x-axis, and cosine is its x-coordinate, sine its y. Now the functions are defined for every angle, positive or negative, and the signs fall out of the quadrant — cosine negative on the left half of the circle, sine negative on the bottom. Tangent is the slope of that radius line, which is why it blows up where the line goes vertical: tan(89°) = 57.3, tan(89.9°) = 573, tan(89.99°) = 5730. Anything in your system proportional to a tangent — a lead screw's wedge action, a gimbal near lock, a ray-trace near grazing — inherits that blow-up.

The behavior worth internalizing:

They oscillate because of their derivatives. Sine's rate of change is cosine, cosine's is minus sine; differentiate twice and you're back where you started with a minus sign. That closed loop is exactly the differential equation x'' = −ω²·x, so a mass on a spring, an LC tank, and a torsion pendulum all trace sines whether anyone asked for trigonometry or not.
Cosine is just shifted sine. cos(θ) = sin(θ + 90°). One family of curves, two starting points. This is also why a·cos(ωt) + b·sin(ωt) is always a single sinusoid R·sin(ωt + φ) with R = √(a² + b²) — two same-frequency sines can interfere but never change frequency.
Symmetry does real work. Sine is odd (sin(−θ) = −sin(θ)), cosine is even (cos(−θ) = cos(θ)). Fourier analysis sorts every signal into those two bins, which is the whole reason the sine/cosine pair, and not some other waveform, is the basis set.

The mistakes people make, in observed order:

Radian/degree mode. sin() in Excel, Python, C, and MATLAB takes radians. A 35° angle fed in raw returns sin(35 rad) = −0.428 — a plausible-looking, sign-flipped, completely wrong number that sails through review. Convert first, and remember the calculus identities (d/dθ sin = cos, sin θ ≈ θ) are radian-only facts; in degrees they pick up factors of π/180.
Trusting the inverse. sin(66.67°) = sin(113.33°) = 0.9182. Hand 0.9182 to asin and you get 66.67° every time, even when the geometry wants the obtuse answer. Same trap with atan: it can't tell the first quadrant from the third. Use atan2(y, x), which takes both signs and returns the true quadrant — it exists in every language because enough people got burned.
Swapping sine and cosine at the angle reference. Component formulas are always quoted "from the horizontal" or "from the axis," and half the field's sign errors come from measuring from the other one. Check with a limit: at θ = 0 the cable pulls straight along its line, so the along component must be the one reading 1, and that's cosine.
Stretching the small-angle approximation. The error term is θ³/6, which is why it dies so fast below 10° and grows so fast past 20°. A flight-dynamics model linearized at 5° of sideslip does not get to be quoted at 25°.

Limit of validity: none in the usual sense — the functions are exact and defined everywhere (tangent excepted at odd multiples of 90°). The approximations built on top of them, and the finite word length of whatever computes them, are where validity questions actually live. Near a zero crossing, 1 − cos(θ) computed naively loses most of its digits; the versions 2·sin²(θ/2) exist for that reason.

History (who derived it and when, told as a short story with inline [n] citations)

Trigonometry was invented to point at the sky, and for its first six hundred years nobody used sine at all. Hipparchus of Nicaea compiled the first known trigonometric table around 140 BC — a table of chords, the straight-line distance across a circular arc — and earned the title "father of trigonometry" for it ^[1]^[2]. Ptolemy's Almagest industrialized the idea: a chord table from half a degree to 180° in half-degree steps, computed by inscribing polygons in a circle, good enough to run Greek astronomy for centuries ^[1]^[2].

The half-chord — our sine — is an Indian invention. Aryabhata, writing around 500 AD, tabulated half-chords at 3.75° intervals and called the quantity jya ^[1]^[2]. The name then took the strangest walk in mathematical vocabulary: Arabic astronomers transliterated jya as jiba, a made-up sound with no Arabic meaning; later copyists read it as jaib, a real word meaning fold or bay; and twelfth-century Latin translators, seeing jaib, faithfully rendered it as sinus — Latin for fold ^[1]^[2]. Sine is a translation of a misreading of a transliteration, and it stuck.

The full six-function toolkit is due largely to Abu al-Wafa al-Buzjani (940–998), working in Baghdad. He put tangent to systematic use, introduced secant and cosecant, studied how all six functions relate, and computed sine tables at 15-arcminute steps accurate to the equivalent of eight decimal places — where Ptolemy had managed three ^[1]^[3]. The modern names came later and from an unlikely place: the Danish mathematician Thomas Fincke coined "tangent" and "secant" in his 1583 Geometria rotundi ^[2]^[4].

The last conceptual step was Euler's. His 1748 Introductio in analysin infinitorum recast sine and cosine — until then lines attached to circles and looked up in tables — as functions of a variable, tied together by e^(iθ) = cos(θ) + i·sin(θ) ^[1]^[2]. That identity is why an EE can multiply phasors instead of wrestling angle-sum identities, and it is the version of trigonometry every engineer actually uses.

Related tools (bullet list of HE calculator slugs that use or neighbor this topic, as /tools/ links)

/tools/convert-angle — degrees, radians, and mils before anything hits sin()
/tools/rms-peak — the 1/√2 sine-wave relationships between RMS, peak, and peak-to-peak
/tools/series-rlc-impedance — magnitude and phase angle, which is trigonometry wearing an impedance triangle
/tools/projectile-range — launch-angle sines doing trajectory work
/tools/pendulum-period — the small-angle approximation earning its keep
/tools/oblique-shock — the θ-β-M relation, tangent functions all the way down

Sources

Written by HE in our own words from the cited sources — engineering judgment included, your stamp still required. All entries →