To find the value of \(\displaystyle{\sqrt[{{3}}]{{{\cos{{40}}}^{{\circ}}}}}\) .

daurialebraslmc

daurialebraslmc

Answered question

2022-04-11

To find the value of cos403 . Wolfram says it is a root of the cubic 8x36x+1 but how to prove that. Note that
(ei2π)19=(cos2π+isin2π)19=(1)19
Can anything be concluded from this?

Answer & Explanation

CyncgotoCancey1k6

CyncgotoCancey1k6

Beginner2022-04-12Added 6 answers

Hint:
cos120=12
cos3x=4cos3x3cosx
phoenixtreeaung

phoenixtreeaung

Beginner2022-04-13Added 19 answers

Let x=cos(40)13, then you have x3=cos(40). Now note that
cos(120)=4cos3(40)3cos(40)12=4x93x38x96x3+1=0

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