Complex Numbers · step 10 of 13

Euler’s formula & trigonometric identities

Euler’s formula e^(iθ) = cos θ + i sin θ turns trigonometry into algebra. Multiplying exponentials and comparing real and imaginary parts derives the angle-sum formulas in two lines, and the inverse relations cos θ = (e^(iθ) + e^(−iθ))/2 and sin θ = (e^(iθ) − e^(−iθ))/(2i) convert powers of sines and cosines into multiple angles — exactly what integration needs.

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Method: how to approach it

The order below is what examiners expect to see, and each step carries its own marks.

State Euler’s formulae^(iθ) = cos θ + i sin θ, and e^(−iθ) = cos θ − i sin θ.
Derive sum formulas by multiplyinge^(i(A+B)) = e^(iA)e^(iB); expand both sides and equate real and imaginary parts.
Use the inverse relations for powersWrite cos θ in exponential form, expand with the binomial theorem, and reassemble into cosines of multiple angles.
Equate parts carefullyReal with real, imaginary with imaginary — this step is where the identities actually come out.

Worked example

Derive the angle-sum formulas for sin(A+B) and cos(A+B) using Euler’s formula.

e^(i(A+B)) = cos(A+B) + i sin(A+B).
Also e^(iA)e^(iB) = (cos A + i sin A)(cos B + i sin B).
Expand: cos A cos B + i cos A sin B + i sin A cos B + i² sin A sin B.
= (cos A cos B − sin A sin B) + i(sin A cos B + cos A sin B).

Answer. Equating real parts gives cos(A+B) = cos A cos B − sin A sin B; equating imaginary parts gives sin(A+B) = sin A cos B + cos A sin B.

Where marks get dropped

These are the specific errors that cost credit on euler’s formula & trigonometric identities questions — QED's rubric penalises each of them separately.

Equating a real part with an imaginary part. The two comparisons are separate equations and must not be mixed.
Forgetting the i in the denominator of sin θ = (e^(iθ) − e^(−iθ))/(2i). Without it the formula is wrong by a factor of i.
Using degrees with e^(iθ). The exponential form requires radians.

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Euler’s formula & trigonometric identities — frequently asked questions

How is Euler’s formula proved?

Compare Maclaurin series: e^(iθ) = Σ(iθ)ⁿ/n!, and separating real and imaginary terms gives precisely the series for cos θ and sin θ.

Why does it simplify integration?

Because ∫cos⁴θ dθ is hard directly, but writing cos θ exponentially and expanding turns it into a sum of cos(kθ) terms, each trivial to integrate.

What is the hyperbolic connection?

cos(iθ) = cosh θ and sin(iθ) = i sinh θ. Every trig identity has a hyperbolic counterpart obtained by this substitution.

The rest of Complex Numbers

Arithmetic, the Argand plane, polar form and De Moivre. Each subtopic below has its own method, worked example and mark-losing traps.

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