MathLabs

Problem 5

Prove that cos⁡π7−cos⁡2π7+cos⁡3π7=12\cos\frac{\pi}{7}-\cos\frac{2\pi}{7}+\cos\frac{3\pi}{7}=\frac12.
Step 1 of 5: Use the seventh roots of unity
Previous stepNext step
1+ω+ω2+⋯+ω6=0,ω=cis⁡2π71+\omega+\omega^2+\cdots+\omega^6=0,\qquad\omega=\operatorname{cis}\frac{2\pi}{7}
Detailed analysis

Let ω=cis⁡(2π/7)\omega=\operatorname{cis}(2\pi/7). Then 1,ω,…,ω61,\omega,\ldots,\omega^6 are the seven seventh roots of unity, whose sum is zero.