Maths Olympiad Prep

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HMMT February

Algebra Difficulty 5.1 AIME, harder Find the answer

Express sin⁡10+sin⁡20+sin⁡30+sin⁡40+sin⁡50+sin⁡60+sin⁡70+sin⁡80cos⁡5cos⁡10cos⁡20\frac{\sin 10+\sin 20+\sin 30+\sin 40+\sin 50+\sin 60+\sin 70+\sin 80}{\cos 5 \cos 10 \cos 20} without using trigonometric functions.

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

We will use the identities cos⁡a+cos⁡b=2cos⁡a+b2cos⁡a−b2\cos a+\cos b=2 \cos \frac{a+b}{2} \cos \frac{a-b}{2} and sin⁡a+sin⁡b=\sin a+\sin b= 2sin⁡a+b2cos⁡a+b22 \sin \frac{a+b}{2} \cos \frac{a+b}{2}. The numerator is (sin⁡10+sin⁡80)+(sin⁡20+sin⁡70)+(sin⁡30+sin⁡60)+(sin⁡40+(\sin 10+\sin 80)+(\sin 20+\sin 70)+(\sin 30+\sin 60)+(\sin 40+ sin⁡60)=2sin⁡45(cos⁡35+cos⁡25+cos⁡15+cos⁡35)=2sin⁡45((cos⁡35+cos⁡5)+(cos⁡25+cos⁡15))=\sin 60)=2 \sin 45(\cos 35+\cos 25+\cos 15+\cos 35)=2 \sin 45((\cos 35+\cos 5)+(\cos 25+\cos 15))= 4sin⁡45cos⁡20(cos⁡15+cos⁡5)=8sin⁡45cos⁡20cos⁡10cos⁡54 \sin 45 \cos 20(\cos 15+\cos 5)=8 \sin 45 \cos 20 \cos 10 \cos 5, so the fraction equals 8sin⁡45=428 \sin 45=4 \sqrt{2}.

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