Cos all the way

Geometry Level 3

If cos ⁡ 2 0 ∘ cos ⁡ 4 0 ∘ cos ⁡ 8 0 ∘ = a b \cos 20 ^ \circ \cos 40 ^ \circ \cos 80 ^ \circ = \frac {a}{b} , where a a and b b are positive coprime integers, what is a + b a + b ?


The answer is 9.

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Calvin Lin by Calvin Lin

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3 solutions

cos ⁡ 2 0 o cos ⁡ 4 0 o cos ⁡ 8 0 o \cos20^o\cos40^o\cos80^o

= 1 2 sin ⁡ 2 0 o ( 2 sin ⁡ 2 0 o cos ⁡ 2 0 o ) cos ⁡ 4 0 o cos ⁡ 8 0 o =\frac{1}{2\sin20^o}(2\sin20^o\cos20^o)\cos40^o\cos80^o

= 1 4 sin ⁡ 2 0 o ( 2 sin ⁡ 4 0 o cos ⁡ 4 0 o ) cos ⁡ 8 0 o =\frac{1}{4\sin20^o}(2\sin40^o\cos40^o)\cos80^o

= 1 8 sin ⁡ 2 0 o ( 2 sin ⁡ 8 0 o cos ⁡ 8 0 o ) =\frac{1}{8\sin20^o}(2\sin80^o\cos80^o)

= 1 8 sin ⁡ 2 0 o sin ⁡ 16 0 o =\frac{1}{8\sin20^o}\sin160^o

= sin ⁡ ( 18 0 o − 2 0 o ) 8 sin ⁡ 2 0 o =\frac{\sin(180^o-20^o)}{8\sin20^o}

= 1 8 =\frac{1}{8}

and 1 + 8 = 9 1+8=9 . What is easier than this?

3 Helpful 0 Interesting 0 Brilliant 0 Confused

good solution..

Takwa Tri Subekti Subekti - 7 years, 3 months ago

goosd

muhammad azam - 7 years, 2 months ago

Wow.. great

Zubair Ahmed - 5 years, 10 months ago
Vishnudatt Gupta
May 2, 2014

1/8 is the answer

1+8=9

0 Helpful 0 Interesting 0 Brilliant 0 Confused
Mukesh G
Dec 16, 2013

cos 20 x cos 40 x cos 80 = 0.125

0.125 = 1/8

a/b = 1/8

a+b =9

0 Helpful 0 Interesting 0 Brilliant 0 Confused

It is a serious breach of problem solving strategy. I just want you to understand that most of the case, they don't allow calculators at exam halls (olympiads). Keep practicing!

Sheikh Asif Imran Shouborno - 7 years, 5 months ago

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