Minimal Polynomial Thing 10

Algebra Level 3

Let p ( x ) p(x) be the monic minimal polynomial of

sin ⁡ ( π 5 ) + cos ⁡ ( π 5 ) + tan ⁡ ( π 5 ) + cot ⁡ ( π 5 ) + csc ⁡ ( π 5 ) + sec ⁡ ( π 5 ) . \sin \left(\frac{\pi }{5}\right)+\cos \left(\frac{\pi }{5}\right)+\tan \left(\frac{\pi }{5}\right)+\cot \left(\frac{\pi }{5}\right)+\csc \left(\frac{\pi }{5}\right)+\sec \left(\frac{\pi }{5}\right).

Submit the sum of the coefficients of p ( x ) p(x) .


The answer is -340.25.

Christopher Criscitiello by Christopher Criscitiello , 25, USA

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1 solution

Here is a bash:

Using this problem ,

sin ⁡ ( π 5 ) + cos ⁡ ( π 5 ) + tan ⁡ ( π 5 ) + cot ⁡ ( π 5 ) + csc ⁡ ( π 5 ) + sec ⁡ ( π 5 ) = ( 5 − 1 ) 2 − 1 5 − 1 + ( 5 − 1 ) 2 − 1 + 1 1 5 − 1 + 1 ( 5 − 1 ) 2 − 1 5 − 1 + 1 5 − 1 + 1 ( 5 − 1 ) 2 − 1 = 1 4 ( 5 5 + 62 5 + 170 − 3 ) \sin\left(\frac{\pi}{5}\right)+\cos\left(\frac{\pi}{5}\right)+\tan\left(\frac{\pi}{5}\right)+\cot\left(\frac{\pi}{5}\right)+\csc\left(\frac{\pi}{5}\right)+\sec\left(\frac{\pi}{5}\right)=\frac{\sqrt{\left(\sqrt{5}-1\right)^2-1}}{\sqrt{5}-1}+\sqrt{ \left(\sqrt{5}-1\right)^2-1}+\frac{1}{\frac{1}{\sqrt{5}-1}}+\frac{1}{\frac{\sqrt{\left(\sqrt{5}-1\right)^2-1}}{\sqrt{5}-1}}+\frac{1}{\sqrt{5}-1}+\frac{1}{\sqrt{\left(\sqrt{5}-1\right)^2-1}}=\frac{1}{4}\left(5\sqrt{5}+\sqrt{62\sqrt{5}+170}-3\right)

Then doing as we did in this problem , we get the answer.

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