Question Number 185320 by mathlove last updated on 20/Jan/23
Answered by SEKRET last updated on 20/Jan/23
$$\:\boldsymbol{\mathrm{sin}}\left(\mathrm{2}\boldsymbol{\mathrm{a}}\right)=\mathrm{2}\boldsymbol{\mathrm{sin}}\left(\boldsymbol{\mathrm{a}}\right)\:\boldsymbol{\mathrm{cos}}\left(\boldsymbol{\mathrm{a}}\right) \\ $$
Answered by SEKRET last updated on 20/Jan/23
![sin((π/2^(10) ))βcos((π/2^(10) ))βcos((π/2^9 ))β........βcos((π/2^2 ))= A 2βsin((π/2^(10) ))βcos((π/2^(10) ))βcos((π/2^9 ))β....βcos((π/2^2 ))=2A sin((π/2^9 ))βcos((π/2^9 ))βcos((π/2^8 ))β....βcos((π/2^2 ))= 2A 2sin((π/2^9 ))βcos((π/2^9 ))βcos((π/2^8 ))....βcos((π/2^2 ))=2^2 βA 2βsin((π/2^8 ))βcos((π/2^8 )).....βcos((π/2^2 ))= 2^3 A ........... ................. ........................ 2sin((π/2^2 ))βcos((π/2^2 ))=2^9 βA sin((π/2))= 2^9 βA [((A= (1/2^9 ) )),(( A= (1/(512)) )) ] ABDULAZIZ ABDUVALIYEV](https://www.tinkutara.com/question/Q185322.png)
$$\:\:\boldsymbol{\mathrm{sin}}\left(\frac{\boldsymbol{\pi}}{\mathrm{2}^{\mathrm{10}} }\right)\centerdot\boldsymbol{\mathrm{cos}}\left(\frac{\boldsymbol{\pi}}{\mathrm{2}^{\mathrm{10}} }\right)\centerdot\boldsymbol{\mathrm{cos}}\left(\frac{\boldsymbol{\pi}}{\mathrm{2}^{\mathrm{9}} }\right)\centerdot……..\centerdot\boldsymbol{\mathrm{cos}}\left(\frac{\boldsymbol{\pi}}{\mathrm{2}^{\mathrm{2}} }\right)=\:\boldsymbol{\mathrm{A}} \\ $$$$\:\mathrm{2}\centerdot\boldsymbol{\mathrm{sin}}\left(\frac{\boldsymbol{\pi}}{\mathrm{2}^{\mathrm{10}} }\right)\centerdot\boldsymbol{\mathrm{cos}}\left(\frac{\boldsymbol{\pi}}{\mathrm{2}^{\mathrm{10}} }\right)\centerdot\boldsymbol{\mathrm{cos}}\left(\frac{\boldsymbol{\pi}}{\mathrm{2}^{\mathrm{9}} }\right)\centerdot….\centerdot\boldsymbol{\mathrm{cos}}\left(\frac{\boldsymbol{\pi}}{\mathrm{2}^{\mathrm{2}} }\right)=\mathrm{2}\boldsymbol{{A}} \\ $$$$\:\:\boldsymbol{\mathrm{sin}}\left(\frac{\boldsymbol{\pi}}{\mathrm{2}^{\mathrm{9}} }\right)\centerdot\boldsymbol{\mathrm{cos}}\left(\frac{\boldsymbol{\pi}}{\mathrm{2}^{\mathrm{9}} }\right)\centerdot\boldsymbol{\mathrm{cos}}\left(\frac{\boldsymbol{\pi}}{\mathrm{2}^{\mathrm{8}} }\right)\centerdot….\centerdot\boldsymbol{\mathrm{cos}}\left(\frac{\boldsymbol{\pi}}{\mathrm{2}^{\mathrm{2}} }\right)=\:\mathrm{2}\boldsymbol{\mathrm{A}} \\ $$$$\:\mathrm{2}\boldsymbol{\mathrm{sin}}\left(\frac{\boldsymbol{\pi}}{\mathrm{2}^{\mathrm{9}} }\right)\centerdot\boldsymbol{\mathrm{cos}}\left(\frac{\boldsymbol{\pi}}{\mathrm{2}^{\mathrm{9}} }\right)\centerdot\boldsymbol{\mathrm{cos}}\left(\frac{\boldsymbol{\pi}}{\mathrm{2}^{\mathrm{8}} }\right)….\centerdot\boldsymbol{\mathrm{cos}}\left(\frac{\boldsymbol{\pi}}{\mathrm{2}^{\mathrm{2}} }\right)=\mathrm{2}^{\mathrm{2}} \centerdot\boldsymbol{\mathrm{A}} \\ $$$$\:\:\mathrm{2}\centerdot\boldsymbol{\mathrm{sin}}\left(\frac{\boldsymbol{\pi}}{\mathrm{2}^{\mathrm{8}} }\right)\centerdot\boldsymbol{\mathrm{cos}}\left(\frac{\boldsymbol{\pi}}{\mathrm{2}^{\mathrm{8}} }\right)…..\centerdot\boldsymbol{\mathrm{cos}}\left(\frac{\boldsymbol{\pi}}{\mathrm{2}^{\mathrm{2}} }\right)=\:\mathrm{2}^{\mathrm{3}} \boldsymbol{\mathrm{A}} \\ $$$$……….. \\ $$$$…………….. \\ $$$$…………………… \\ $$$$\:\:\:\:\mathrm{2}\boldsymbol{\mathrm{sin}}\left(\frac{\boldsymbol{\pi}}{\mathrm{2}^{\mathrm{2}} }\right)\centerdot\boldsymbol{\mathrm{cos}}\left(\frac{\boldsymbol{\pi}}{\mathrm{2}^{\mathrm{2}} }\right)=\mathrm{2}^{\mathrm{9}} \centerdot\boldsymbol{\mathrm{A}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\boldsymbol{\mathrm{sin}}\left(\frac{\boldsymbol{\pi}}{\mathrm{2}}\right)=\:\mathrm{2}^{\mathrm{9}} \centerdot\boldsymbol{\mathrm{A}}\:\:\:\:\:\:\:\:\:\begin{bmatrix}{\boldsymbol{\mathrm{A}}=\:\frac{\mathrm{1}}{\mathrm{2}^{\mathrm{9}} }\:\:\:\:\:}\\{\:\:\boldsymbol{{A}}=\:\:\frac{\mathrm{1}}{\mathrm{512}}\:\:}\end{bmatrix} \\ $$$$\:\boldsymbol{{ABDULAZIZ}}\:\:\:\boldsymbol{{ABDUVALIYEV}} \\ $$