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Question Number 185320 by mathlove last updated on 20/Jan/23

Answered by SEKRET last updated on 20/Jan/23

 sin(2a)=2sin(a) cos(a)

$$\:\boldsymbol{\mathrm{sin}}\left(\mathrm{2}\boldsymbol{\mathrm{a}}\right)=\mathrm{2}\boldsymbol{\mathrm{sin}}\left(\boldsymbol{\mathrm{a}}\right)\:\boldsymbol{\mathrm{cos}}\left(\boldsymbol{\mathrm{a}}\right) \\ $$

Answered by SEKRET last updated on 20/Jan/23

  sin((𝛑/2^(10) ))βˆ™cos((𝛑/2^(10) ))βˆ™cos((𝛑/2^9 ))βˆ™........βˆ™cos((𝛑/2^2 ))= A   2βˆ™sin((𝛑/2^(10) ))βˆ™cos((𝛑/2^(10) ))βˆ™cos((𝛑/2^9 ))βˆ™....βˆ™cos((𝛑/2^2 ))=2A    sin((𝛑/2^9 ))βˆ™cos((𝛑/2^9 ))βˆ™cos((𝛑/2^8 ))βˆ™....βˆ™cos((𝛑/2^2 ))= 2A   2sin((𝛑/2^9 ))βˆ™cos((𝛑/2^9 ))βˆ™cos((𝛑/2^8 ))....βˆ™cos((𝛑/2^2 ))=2^2 βˆ™A    2βˆ™sin((𝛑/2^8 ))βˆ™cos((𝛑/2^8 )).....βˆ™cos((𝛑/2^2 ))= 2^3 A  ...........  .................  ........................      2sin((𝛑/2^2 ))βˆ™cos((𝛑/2^2 ))=2^9 βˆ™A             sin((𝛑/2))= 2^9 βˆ™A          [((A= (1/2^9 )     )),((  A=  (1/(512))  )) ]   ABDULAZIZ   ABDUVALIYEV

$$\:\:\boldsymbol{\mathrm{sin}}\left(\frac{\boldsymbol{\pi}}{\mathrm{2}^{\mathrm{10}} }\right)\centerdot\boldsymbol{\mathrm{cos}}\left(\frac{\boldsymbol{\pi}}{\mathrm{2}^{\mathrm{10}} }\right)\centerdot\boldsymbol{\mathrm{cos}}\left(\frac{\boldsymbol{\pi}}{\mathrm{2}^{\mathrm{9}} }\right)\centerdot........\centerdot\boldsymbol{\mathrm{cos}}\left(\frac{\boldsymbol{\pi}}{\mathrm{2}^{\mathrm{2}} }\right)=\:\boldsymbol{\mathrm{A}} \\ $$$$\:\mathrm{2}\centerdot\boldsymbol{\mathrm{sin}}\left(\frac{\boldsymbol{\pi}}{\mathrm{2}^{\mathrm{10}} }\right)\centerdot\boldsymbol{\mathrm{cos}}\left(\frac{\boldsymbol{\pi}}{\mathrm{2}^{\mathrm{10}} }\right)\centerdot\boldsymbol{\mathrm{cos}}\left(\frac{\boldsymbol{\pi}}{\mathrm{2}^{\mathrm{9}} }\right)\centerdot....\centerdot\boldsymbol{\mathrm{cos}}\left(\frac{\boldsymbol{\pi}}{\mathrm{2}^{\mathrm{2}} }\right)=\mathrm{2}\boldsymbol{{A}} \\ $$$$\:\:\boldsymbol{\mathrm{sin}}\left(\frac{\boldsymbol{\pi}}{\mathrm{2}^{\mathrm{9}} }\right)\centerdot\boldsymbol{\mathrm{cos}}\left(\frac{\boldsymbol{\pi}}{\mathrm{2}^{\mathrm{9}} }\right)\centerdot\boldsymbol{\mathrm{cos}}\left(\frac{\boldsymbol{\pi}}{\mathrm{2}^{\mathrm{8}} }\right)\centerdot....\centerdot\boldsymbol{\mathrm{cos}}\left(\frac{\boldsymbol{\pi}}{\mathrm{2}^{\mathrm{2}} }\right)=\:\mathrm{2}\boldsymbol{\mathrm{A}} \\ $$$$\:\mathrm{2}\boldsymbol{\mathrm{sin}}\left(\frac{\boldsymbol{\pi}}{\mathrm{2}^{\mathrm{9}} }\right)\centerdot\boldsymbol{\mathrm{cos}}\left(\frac{\boldsymbol{\pi}}{\mathrm{2}^{\mathrm{9}} }\right)\centerdot\boldsymbol{\mathrm{cos}}\left(\frac{\boldsymbol{\pi}}{\mathrm{2}^{\mathrm{8}} }\right)....\centerdot\boldsymbol{\mathrm{cos}}\left(\frac{\boldsymbol{\pi}}{\mathrm{2}^{\mathrm{2}} }\right)=\mathrm{2}^{\mathrm{2}} \centerdot\boldsymbol{\mathrm{A}} \\ $$$$\:\:\mathrm{2}\centerdot\boldsymbol{\mathrm{sin}}\left(\frac{\boldsymbol{\pi}}{\mathrm{2}^{\mathrm{8}} }\right)\centerdot\boldsymbol{\mathrm{cos}}\left(\frac{\boldsymbol{\pi}}{\mathrm{2}^{\mathrm{8}} }\right).....\centerdot\boldsymbol{\mathrm{cos}}\left(\frac{\boldsymbol{\pi}}{\mathrm{2}^{\mathrm{2}} }\right)=\:\mathrm{2}^{\mathrm{3}} \boldsymbol{\mathrm{A}} \\ $$$$........... \\ $$$$................. \\ $$$$........................ \\ $$$$\:\:\:\:\mathrm{2}\boldsymbol{\mathrm{sin}}\left(\frac{\boldsymbol{\pi}}{\mathrm{2}^{\mathrm{2}} }\right)\centerdot\boldsymbol{\mathrm{cos}}\left(\frac{\boldsymbol{\pi}}{\mathrm{2}^{\mathrm{2}} }\right)=\mathrm{2}^{\mathrm{9}} \centerdot\boldsymbol{\mathrm{A}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\boldsymbol{\mathrm{sin}}\left(\frac{\boldsymbol{\pi}}{\mathrm{2}}\right)=\:\mathrm{2}^{\mathrm{9}} \centerdot\boldsymbol{\mathrm{A}}\:\:\:\:\:\:\:\:\:\begin{bmatrix}{\boldsymbol{\mathrm{A}}=\:\frac{\mathrm{1}}{\mathrm{2}^{\mathrm{9}} }\:\:\:\:\:}\\{\:\:\boldsymbol{{A}}=\:\:\frac{\mathrm{1}}{\mathrm{512}}\:\:}\end{bmatrix} \\ $$$$\:\boldsymbol{{ABDULAZIZ}}\:\:\:\boldsymbol{{ABDUVALIYEV}} \\ $$

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