Trigonometric identities

The formula

sin²θ + cos²θ = 1

The Pythagorean identity: any angle's sine and cosine are the legs of a right triangle with hypotenuse 1. Divide through by cos²θ and you get the sibling 1 + tan²θ = sec²θ for free.

sin(a ± b) = sin a · cos b ± cos a · sin b
cos(a ± b) = cos a · cos b ∓ sin a · sin b

The angle-sum formulas — the parents of nearly everything below. Note the sign flip on the cosine line; that ∓ is where most hand-derivation errors are born.

sin 2θ = 2 sin θ · cos θ
cos 2θ = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2 sin²θ

Double angles: set b = a in the sum formulas. Cosine gets three equivalent forms; pick the one that cancels what you're carrying.

sin²θ = (1 − cos 2θ)/2        cos²θ = (1 + cos 2θ)/2

Power reduction: squaring a sinusoid produces a DC term of one-half plus a tone at twice the frequency. This one line is why the RMS of a sine is A/√2 ≈ 0.707·A, why square-law detectors work, and why a full-wave rectifier's output has ripple at 2f.

cos A · cos B = ½[cos(A−B) + cos(A+B)]
sin A · cos B = ½[sin(A+B) + sin(A−B)]

Product-to-sum — the mixer identities. Multiply two tones and you get exactly two new tones, at the difference and the sum, each at half amplitude. No other frequencies, unless your hardware misbehaves.

sin A + sin B = 2 sin((A+B)/2) · cos((A−B)/2)

Sum-to-product: two close tones added together become one carrier at the average frequency inside a slow envelope — the cosine term runs at half the difference frequency, (f₁−f₂)/2. But watch the factor of two: because |cos| peaks twice per cycle, the amplitude maxima — the beats you hear and count on a strip chart or spectrum — arrive at the full difference f₁−f₂. The envelope is the modulation your accelerometer sees; the beat rate you count is the difference itself.

a·cos θ + b·sin θ = R·cos(θ − φ),   R = √(a² + b²),   φ = atan2(b, a)

Harmonic addition: any in-phase plus quadrature pair collapses to a single sinusoid with one amplitude and one phase. 3 cos θ + 4 sin θ is exactly 5 cos(θ − 53.13°).

e^(jθ) = cos θ + j·sin θ

Euler's formula — the master key. Every identity above falls out of multiplying exponentials and matching real and imaginary parts. Memorize this one; derive the rest.

Where you meet it

RF bench, spectrum analyzer on a mixer output. Drive a mixer with a 100 MHz RF tone and a 110.7 MHz LO and product-to-sum says the output holds exactly 10.7 MHz and 210.7 MHz. Everything else on the screen — LO feedthrough, 2LO−RF, the intermod forest — is the hardware failing to be a pure multiplier. You can't sort spur from product unless you know what the ideal math predicts first.
Vibration lab, two sources close in frequency. A shaker fixture excited near two closely spaced modes, or twin pumps running at 1780 and 1786 rpm, produces the classic beat: sum-to-product turns the pair into a carrier inside a slow envelope, and you count amplitude peaks at the full difference — 6 beats per minute, not 3 — so the beat count reads the rpm split directly. Test engineers who don't recognize the identity chase a "surging" ghost; those who do count the beats and go looking for a second source 6 rpm away.
Signal conditioning and demodulation. Lock-in amplifiers, IQ demodulators, and synchro/resolver math all live on the product identities: multiply by a reference cosine, and the identity guarantees your signal lands at DC plus a 2f term the low-pass filter removes. The harmonic-addition identity is why your I and Q channels reduce to one magnitude and one phase on the display.
Review board, someone simplifies before the room simulates. A vibration response written as a·cos ωt + b·sin ωt becomes one R and one φ on the whiteboard; a squared sinusoid in a power calculation drops its DC term by inspection. The engineer who can do this in marker settles arguments before anyone opens MATLAB.

How it works

The whole catalog hangs off two facts. The Pythagorean identity is the unit circle restated — a point at angle θ sits at (cos θ, sin θ), distance 1 from the origin, and Pythagoras does the rest. The angle-sum formulas are what rotation composition looks like written out in components; every double-angle, half-angle, power-reduction, and product formula is the sum formulas with a substitution. Euler's formula compresses all of it: e^(ja)·e^(jb) = e^(j(a+b)), expand both sides, match real and imaginary parts, and both sum formulas fall out in two lines. This is the practical advice buried in the topic: stop memorizing twenty identities and get fluent at deriving them from one exponential.

What the identities buy you on real hardware:

A multiplier is a frequency shifter. Product-to-sum is not an approximation of mixing — it is mixing. An ideal multiplier produces the sum and difference frequencies at half amplitude and nothing else. Every spur beyond those two is a diagnosis, not a surprise.
Squaring makes DC. Power reduction says sin² averages to exactly ½, which is why RMS of a sine is 0.707·A, why RF power meters can be square-law devices, and why the instantaneous power of a single-phase line pulses at twice the line frequency no matter what the load does.
Two terms are always one phasor. Harmonic addition is the identity form of phasor arithmetic. Any same-frequency combination of sines and cosines, however many stages produced it, is a single amplitude and phase.

The mistakes people actually make:

The cosine sign flip. cos(a+b) carries a minus between its terms. Deriving in a hurry, engineers write cos a cos b + sin a sin b — which is cos(a−b) — and the error survives until a phase comes out 90° wrong on hardware.
Degrees into radian functions. Every math library's sin() takes radians. Feeding it degrees doesn't error out; it returns confident nonsense. The identities are exact in either unit, but only if you commit to one.
atan(b/a) instead of atan2(b, a). The harmonic-addition phase needs the two-argument arctangent. The single-argument version folds two quadrants onto the other two, and the resulting 180° phase error is a classic in resolver and IQ processing code.
Confusing identities with approximations. sin²θ + cos²θ = 1 holds for every θ, real or complex. sin θ ≈ θ does not — it's inside 1% only out to about 14°, and tan θ ≈ θ only to about 10°. Reports regularly cite "the trig identity" when they mean the small-angle approximation, and the reviewer who knows the difference will ask which one your gimbal model used.
Numerical self-sabotage. 1 − cos θ at small θ subtracts two nearly equal numbers and shreds precision in single-precision code. The half-angle identity 1 − cos θ = 2 sin²(θ/2) computes the same quantity with no cancellation. Identities aren't just algebra; they're how you rewrite a formula so floating point can survive it.

Limits of validity: there aren't any, and that's the point. These are identities, not models — exact for every argument, including complex ones (where sin and cos grow without bound but the identities hold anyway). The only things that expire are the approximations people attach to them.

History

The identities are older than the functions. Hipparchus of Nicaea built the first known table of chords around 140 BC, working circle geometry for astronomy without any notion of sine or cosine ^[2]^[3]. Three centuries later Ptolemy's Almagest carried theorems on chords of sums and differences of arcs — the angle-addition formulas in chord clothing — plus the equivalent of the half-angle formula, which is how he filled a chord table in half-degree steps ^[2]^[3]. The sine itself arrived by a different road: Aryabhata's work in India around 500 AD tabulated half-chords, the quantity we now call sine ^[2]^[3].

The best engineering story in the topic is prosthaphaeresis. In the 1580s, decades before logarithms, astronomers needed to multiply large trig values by hand, and the product-to-sum identities were the shortcut: scale two numbers into cosines, look up two angles, add and subtract them, look up two more cosines, average. Multiplication became table lookups and addition. Paul Wittich carried the method to Tycho Brahe's observatory at Uraniborg in 1580, and Brahe's data reduction ran on it; Bürgi, Clavius, and Viète all worked versions of the trick ^[4]^[5]. Napier's logarithms of 1614 made it obsolete ^[4] — but the identity never stopped working, and four centuries later every RF mixer on every bench in Huntsville is running prosthaphaeresis in hardware.

The catalog got its modern spine in 1748, when Euler's Introductio in analysin infinitorum treated sine and cosine as functions of a number rather than lengths in a figure, and published e^(ix) = cos x + i sin x ^[2]^[3]^[6]. After Euler, the identities stopped being a list to memorize and became consequences of one exponential law — which is still the right way to hold them.

Related tools

/tools/rms-peak — the A/√2 conversion is the power-reduction identity doing its job
/tools/series-rlc-impedance — magnitude-and-phase output is harmonic addition as a calculator
/tools/convert-angle — degrees, radians, and the unit mistake in gotcha #2
/tools/oblique-shock — the θ–β–M relation is a workout in tangent and double-angle manipulation

Sources

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