A complete ripoff

Geometry Level 5

sin60+sin610+sin620+sin630++sin6180cos60+cos610+cos620+cos630++cos6180\dfrac{\sin^6{0^{\circ}}+\sin^6{10^{\circ}}+\sin^6{20^{\circ}}+\sin^6{30^{\circ}}+\ldots+\sin^6{180^{\circ}}}{\cos^6{0^{\circ}}+\cos^6{10^{\circ}}+\cos^6{20^{\circ}}+\cos^6{30^{\circ}}+\ldots+\cos^6{180^{\circ}}} \\ \\

If the value of the above expression is equals to ab\frac{a}{b} where aa and bb are coprime positive integers, find the value of bab-a.

Bonus: For the general expression below, can you find a general formula for the answer, bnanb_{n} - a_{n} for all positive integers nn?

sin2n0+sin2n10+sin2n20+sin2n30++sin2n180cos2n0+cos2n10+cos2n20+cos2n30++cos2n180\dfrac{\sin^{2n}{0^{\circ}}+\sin^{2n}{10^{\circ}}+\sin^{2n}{20^{\circ}}+\sin^{2n}{30^{\circ}}+\ldots+\sin^{2n}{180^{\circ}}}{\cos^{2n}{0^{\circ}}+\cos^{2n}{10^{\circ}}+\cos^{2n}{20^{\circ}}+\cos^{2n}{30^{\circ}}+\ldots+\cos^{2n}{180^{\circ}}}

Inspiration.

×

Problem Loading...

Note Loading...

Set Loading...