Exercise 1539Combinatorics › Binomial coefficientsOlympiadAdded 3 Nov 2026

Sum of sines

#complex numbers#binomial#sine

Statement

Show that $\sum_{k=0}^{n}\binom{n}{k}\sin(k\theta)=2^n\cos^n(\theta/2)\sin(n\theta/2)$ .

Hint

Take imaginary part of $(1+e^{i\theta})^n$ .

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Imaginary part. $\sum_k\binom{n}{k}\sin(k\theta)=\operatorname{Im}((1+e^{i\theta})^n)$ .

Compute. $(1+e^{i\theta})^n=2^n\cos^n(\theta/2)e^{in\theta/2}$ .

Imaginary part. $2^n\cos^n(\theta/2)\sin(n\theta/2)$ .

Conclusion. The identity holds.