Evaluate cos² (π/7) + cos² (3π/7) + cos² (5π/7) and sin² (π/7) + sin² (3π/7) + sin² (5π/7)

Saya sengaja buat 2 soal ini menjadi soal terstruktur. Soal ini sebenarnya mudah hanya membutuhkan manipulasi dan rumus-rumus trigonometri. Kita kerjakan cos² (π/7) + cos² (3π/7) + cos² (5π/7) terlebih dahulu.

\displaystyle \cos^2 \left ( \frac{\pi}{7} \right )+\cos^2 \left ( \frac{3\pi}{7} \right )+\cos^2 \left ( \frac{5\pi}{7} \right )\\ =\frac{1+\cos \left ( \frac{2\pi}{7} \right )}{2}+\frac{1+\cos \left ( \frac{6\pi}{7} \right )}{2}+\frac{1+\cos \left ( \frac{10\pi}{7} \right )}{2}\\ =\frac{3+\cos \left ( \frac{2\pi}{7} \right )+\cos \left ( \frac{6\pi}{7} \right )+\cos \left ( \frac{10\pi}{7} \right )}{2}\\=\frac{3+\frac{2\sin \left ( \frac{2\pi}{7} \right )}{2\sin \left ( \frac{2\pi}{7} \right )}\left [ \cos \left ( \frac{2\pi}{7} \right )+\cos \left ( \frac{6\pi}{7} \right )+\cos \left ( \frac{10\pi}{7} \right ) \right ]}{2}\\ =\frac{3+\frac{2\sin\left ( \frac{2\pi}{7} \right )\cos\left ( \frac{2\pi}{7} \right )+2\sin\left ( \frac{2\pi}{7} \right )\cos\left ( \frac{6\pi}{7} \right )+2\sin\left ( \frac{2\pi}{7} \right )\cos\left ( \frac{10\pi}{7} \right )}{2\sin\left ( \frac{2\pi}{7} \right )}}{2}\\=\frac{3+\frac{\sin \left ( \frac{4\pi}{7} \right )+\sin \left ( \frac{8\pi}{7} \right )-\sin \left ( \frac{4\pi}{7} \right )+\sin \left ( \frac{12\pi}{7} \right )-\sin \left ( \frac{8\pi}{7} \right )}{2\sin \left ( \frac{2\pi}{7} \right )}}{2}\\ =\frac{3+\frac{\sin \left ( 2\pi-\frac{12\pi}{7} \right )}{2\sin \left ( \frac{2\pi}{7} \right )}}{2}\\=\frac{3-\frac{\sin \left ( \frac{2\pi}{7} \right )}{2\sin \left ( \frac{2\pi}{7} \right )}}{2}\\ =\frac{5}{4}\\ =1,25

 

Sekarang kita kerjakan sin² (π/7) + sin² (3π/7) + sin² (5π/7).

\displaystyle \sin^2 \left ( \frac{\pi}{7} \right )+\sin^2 \left ( \frac{3\pi}{7} \right )+\sin^2 \left ( \frac{5\pi}{7} \right )\\ =\frac{1-\cos \left ( \frac{2\pi}{7} \right )}{2}+\frac{1-\cos \left ( \frac{6\pi}{7} \right )}{2}+\frac{1-\cos \left ( \frac{10\pi}{7} \right )}{2}\\ =\frac{3-\left [ \cos \left ( \frac{2\pi}{7} \right )+\cos \left ( \frac{6\pi}{7} \right )+\cos \left ( \frac{10\pi}{7} \right ) \right ]}{2}\\=\frac{3-\left ( -\frac{1}{2} \right )}{2}\\ =\frac{7}{4}\\ =1,75

 

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