Angle Addition Formulae

Whenever two waves combine — two musical notes blending, radio signals overlapping, the alternating current humming in your walls — the result depends on adding their angles. The formulae on this page are the tool that predicts what you get, and they start from a surprisingly sneaky little puzzle.

Here is a question that looks harmless and hides a trap. You know \sin 30^\circ = \tfrac12 and \sin 45^\circ = \tfrac{\sqrt2}{2}\approx 0.707. So what is \sin 75^\circ? Since 75 = 30 + 45, surely you just add the answers?

\sin 75^\circ \overset{?}{=} \sin 30^\circ + \sin 45^\circ = 0.5 + 0.707 = 1.207.

That cannot be right — sine never climbs above 1. The tempting move “\sin(A+B) = \sin A + \sin B” is wrong. The true answer is more subtle, and the tool that delivers it is one of the most useful gadgets in all of trigonometry: the angle addition (compound-angle) formulae. They give the sine and cosine of a sum or difference of angles in terms of the sines and cosines of the parts — and they unlock exact values, a whole family of identities, and the mathematics of combining waves.

\sin(A + B) = \sin A\cos B + \cos A\sin B \cos(A + B) = \cos A\cos B - \sin A\sin B

The difference versions just flip the middle sign:

\sin(A - B) = \sin A\cos B - \cos A\sin B \cos(A - B) = \cos A\cos B + \sin A\sin B

Notice the pattern: \sin mixes \sin\cos with a matching sign, while \cos keeps each function with itself and flips the sign (a plus on the left becomes a minus on the right).

Why they are useful

Their real power is finding exact values of brand-new angles by splitting them into two angles you already know. For example 75^\circ = 45^\circ + 30^\circ, so

\sin 75^\circ = \sin 45^\circ\cos 30^\circ + \cos 45^\circ\sin 30^\circ = \tfrac{\sqrt2}{2}\cdot\tfrac{\sqrt3}{2} + \tfrac{\sqrt2}{2}\cdot\tfrac12 = \tfrac{\sqrt6 + \sqrt2}{4} \approx 0.966.

Comfortably below 1, exactly as it must be — and a world away from the bogus 1.207.

For any two angles A and B: Splitting an angle as a sum or difference of special angles lets you build exact values for new angles such as 15^\circ (45^\circ - 30^\circ) and 75^\circ (45^\circ + 30^\circ).

Adding angles at a vertex

Placing two angles A and B next to each other at a single vertex gives the combined angle A + B — the geometric idea the formulae capture. The formulae are exactly the bookkeeping for how the sine and cosine of that combined angle depend on the two pieces.

Worked example 1 — an exact value for \cos 15^\circ

Split 15^\circ = 45^\circ - 30^\circ and use the difference formula for cosine (remember: cosine's difference formula carries a plus):

\cos 15^\circ = \cos(45^\circ - 30^\circ) = \cos 45^\circ\cos 30^\circ + \sin 45^\circ\sin 30^\circ. = \tfrac{\sqrt2}{2}\cdot\tfrac{\sqrt3}{2} + \tfrac{\sqrt2}{2}\cdot\tfrac12 = \tfrac{\sqrt6}{4} + \tfrac{\sqrt2}{4} = \tfrac{\sqrt6 + \sqrt2}{4} \approx 0.966.

Interesting: \cos 15^\circ = \sin 75^\circ. That is no coincidence — 15^\circ and 75^\circ add to 90^\circ, so they are complementary, and cosine of one always equals sine of the other.

Worked example 2 — the double-angle formulae fall right out

The single most productive trick with these formulae is to set B = A. Then a “sum of two angles” becomes “twice one angle”, and you read off the double-angle formulae for free. Start with sine:

\sin 2A = \sin(A + A) = \sin A\cos A + \cos A\sin A = 2\sin A\cos A.

Now cosine:

\cos 2A = \cos(A + A) = \cos A\cos A - \sin A\sin A = \cos^2 A - \sin^2 A.

And because \sin^2 A + \cos^2 A = 1, that last line can be rewritten two more handy ways — \cos 2A = 2\cos^2 A - 1 = 1 - 2\sin^2 A. Three faces of one identity, all born from a single substitution.

Divide the sine formula by the cosine formula and simplify (top and bottom by \cos A\cos B) and you get the compound-angle formula for tangent:

\tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A\tan B}.

Set B = A again and it collapses to \tan 2A = \dfrac{2\tan A}{1 - \tan^2 A}. Everything in this whole topic is really the two original formulae wearing different hats.

Worked example 3 — simplifying an expression

The formulae work backwards too: spot the pattern and collapse a long expression into a short one. Take

\sin 50^\circ\cos 20^\circ - \cos 50^\circ\sin 20^\circ.

This is exactly the shape of \sin A\cos B - \cos A\sin B with A = 50^\circ and B = 20^\circ, so it folds up into

\sin(50^\circ - 20^\circ) = \sin 30^\circ = \tfrac12.

Learning to read the formulae in both directions — expand and collapse — is what makes them powerful when you meet integration and proofs later on.

This is one of the most tempting — and most punished — mistakes in all of mathematics. Trig functions do not distribute over addition:

\sin(A + B) \ne \sin A + \sin B, \qquad \cos(A + B) \ne \cos A + \cos B.

You saw the proof at the top: \sin 30^\circ + \sin 45^\circ \approx 1.207, which is impossible for a sine. You must use the full formula. And mind the sign pattern while you are at it: \cos(A+B) has a minus in the middle (\cos A\cos B - \sin A\sin B) even though the left-hand side had a plus — the signs “flip” for cosine. Sine keeps the sign it was given; cosine reverses it.

The angle addition formulae are the reason two sound waves of the same frequency add up into a single wave of that same frequency — just taller, shorter, or shifted along. Write two waves as a\sin\theta and b\cos\theta; the formulae let you combine them into one clean R\sin(\theta + \varphi) with a new amplitude R and a phase shift \varphi.

Flip \varphi to 180^\circ and the second wave becomes the exact upside-down twin of the first, so they cancel to silence — that is precisely what noise-cancelling headphones do, generating an “anti-noise” wave to erase the hum around you. The same algebra runs through AC electricity and radio: engineers analyse mains voltage and radio signals by adding and shifting sine waves all day long. These formulae are also the gateway to the double- and half-angle results, and — one step further — to Fourier analysis, the astonishing idea that any signal is a sum of sine waves. That single idea is what makes MP3s, JPEGs, and very nearly everything digital possible.

See it explained