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Geometry Level 4

( 1 2 + cos ⁡ ( π 20 ) ) ( 1 2 + cos ⁡ ( 3 π 20 ) ) ( 1 2 + cos ⁡ ( 9 π 20 ) ) ( 1 2 + cos ⁡ ( 27 π 20 ) ) = ? \large \left( \frac12 + \cos\left(\frac\pi{20}\right)\right) \left( \frac12 + \cos\left(\frac{3\pi}{20}\right)\right) \left( \frac12 + \cos\left(\frac{9\pi}{20}\right)\right) \left( \frac12 + \cos\left(\frac{27\pi}{20}\right)\right) = \ ?

Give your answer to 4 decimal places.

This is a very very beautiful problem. This is a part of Trigo Plus! .


The answer is 0.0625.

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Ravi Dwivedi by Ravi Dwivedi , 24, India

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1 solution

Ravi Dwivedi
Jul 13, 2015

1 + 2 cos ⁡ 2 x = 1 + 2 ( 1 − 2 sin ⁡ 2 x ) 1+2 \cos 2x =1+2(1-2 \sin^2 x) = 3 − 4 sin ⁡ 2 x = sin ⁡ 3 x sin ⁡ x =3-4 \sin^2 x= \large \frac{\sin 3x}{\sin x}

Given product becomes P = ( 1 2 + cos ⁡ π 20 ) ( 1 2 + cos ⁡ 3 π 20 ) ( 1 2 + cos ⁡ 9 π 20 ) ( 1 2 + cos ⁡ 27 π 20 ) P=(\frac{1}{2}+ \cos \frac{\pi}{20})(\frac{1}{2}+ \cos \frac{3\pi}{20})(\frac{1}{2}+ \cos \frac{9\pi}{20})(\frac{1}{2}+ \cos \frac{27\pi}{20})\\ = ⋅ 1 2 4 sin ⁡ 3 π 20 sin ⁡ π 20 sin ⁡ 9 π 20 sin ⁡ 3 π 20 sin ⁡ 27 π 20 sin ⁡ 9 π 20 sin ⁡ 81 π 20 sin ⁡ 27 π 20 = \cdot \frac{1}{2^4} \frac{\sin \frac{3\pi}{20}}{\sin \frac{\pi}{20}} \frac{\sin \frac{9\pi}{20}}{\sin \frac{3\pi}{20}} \frac{\sin \frac{27\pi}{20}}{\sin \frac{9\pi}{20}} \frac{\sin \frac{81\pi}{20}}{\sin \frac{27\pi}{20}} = 1 2 4 sin ⁡ 81 π 20 sin ⁡ π 20 =\frac{1}{2^4}\frac{\sin \frac{81\pi}{20}}{\sin \frac{\pi}{20}}

Now sin ⁡ 81 π 20 = sin ⁡ ( 4 π + π 20 ) = sin ⁡ π 20 \sin \frac{81\pi}{20} = \sin (4\pi + \frac{\pi}{20}) = \sin\frac{\pi}{20}

Putting back we get P = 1 2 4 = 0.0625 P= \frac{1}{2^4}= \boxed {0.0625}

4 Helpful 0 Interesting 0 Brilliant 0 Confused

Moderator note:

Interesting observation. How can one solve this problem without already knowing this identity?

Note that in your denominator, the first sin ⁡ \sin term should be sin ⁡ π 20 \sin \frac{\pi}{20} instead of sin ⁡ 9 π 20 \sin \frac{ 9 \pi } { 20} .

very good solution upvoted....but when you have solved this how did you know or how it comes in your mind to use which identity..please help me i 'm struggling in trignometry

Harshi Singh - 5 years, 8 months ago

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Shobhit Singh - 5 years, 11 months ago

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