Sines everywhere 2

Geometry Level 4

sin ⁡ ( x ) − 2 sin ⁡ ( 2 x ) − sin ⁡ ( 3 x ) = 2 2 \large \sin(x)-2\sin(2x)-\sin(3x)=2\sqrt{2}

Find the sum of all real x ∈ [ 0 ; 2 π ] x\in [0;2\pi] satisfying the equation above, submit your answer to 2 decimal places.

Note : If you think there is no root, submit 100.00 as your answer.


The answer is 100.00.

P C by P C , 20, Vietnam

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2 solutions

Aaghaz Mahajan
Jul 8, 2018

For a Non-Graphical approach, use sum to product for converting sin(3x) - sin(x) into 2cos(2x)sin(x)..........
Then, shift all trigonometric ratios to one side, and simply apply Cauchy-Schwarz........!!! We see that the equality condition is not satisfied..........!!!! @P C Is my approach correct???

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@P C Sir, please reply.......

Aaghaz Mahajan - 2 years, 11 months ago
Sahil Silare
Sep 21, 2016

From graph we have, So as y = 2 2 y=2\sqrt{2} doesn't intersect the function there's no solution to it.

1 Helpful 0 Interesting 0 Brilliant 0 Confused

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