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Question Number 216202 by mahdipoor last updated on 29/Jan/25
prove : sin(a+b)=sin(a)cos(b)+sin(b)cos(a)
$${prove}\::\: \\ $$$${sin}\left({a}+{b}\right)={sin}\left({a}\right){cos}\left({b}\right)+{sin}\left({b}\right){cos}\left({a}\right) \\ $$
Answered by dionigi last updated on 30/Jan/25
e^(i (a+b) ) = cos(a+b) + i sin(a+b) e^(i (a+b) ) = e^(ia) e^(ib) e^(i (a+b) ) = (cos a + i sin a) (cos b + i sin b) e^(i (a+b) ) = cos a cos b − sin a sin b + i sin a cos b + i cos a sin b spliting real and imaginary parts of first and last line cos(a+b) = cos a cos b −sin a sin b sin(a+b) = sin a cos b + cos a sin b
$${e}^{{i}\:\left({a}+{b}\right)\:} =\:\mathrm{cos}\left({a}+{b}\right)\:+\:{i}\:\mathrm{sin}\left({a}+{b}\right) \\ $$$${e}^{{i}\:\left({a}+{b}\right)\:} =\:{e}^{{ia}} \:{e}^{{ib}} \: \\ $$$${e}^{{i}\:\left({a}+{b}\right)\:} =\:\left(\mathrm{cos}\:{a}\:+\:{i}\:\mathrm{sin}\:{a}\right)\:\left(\mathrm{cos}\:{b}\:+\:{i}\:\mathrm{sin}\:{b}\right) \\ $$$${e}^{{i}\:\left({a}+{b}\right)\:} =\:\mathrm{cos}\:{a}\:\mathrm{cos}\:{b}\:−\:\mathrm{sin}\:{a}\:\mathrm{sin}\:{b}\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:+\:{i}\:\mathrm{sin}\:{a}\:\mathrm{cos}\:{b}\:+\:{i}\:\mathrm{cos}\:{a}\:\mathrm{sin}\:{b} \\ $$$${spliting}\:{real}\:{and}\:{imaginary}\:{parts} \\ $$$${of}\:{first}\:{and}\:{last}\:{line} \\ $$$$\mathrm{cos}\left({a}+{b}\right)\:=\:\mathrm{cos}\:{a}\:\mathrm{cos}\:{b}\:−\mathrm{sin}\:{a}\:\mathrm{sin}\:{b} \\ $$$$\mathrm{sin}\left({a}+{b}\right)\:=\:\mathrm{sin}\:{a}\:\mathrm{cos}\:{b}\:+\:\mathrm{cos}\:{a}\:\mathrm{sin}\:{b} \\ $$
Answered by mehdee7396 last updated on 30/Jan/25
EC=EF×Cosα & EB=AE×Sinα Sin(α+β)=((AD)/(AF))=((EC)/(AF))+((BE)/(AF)) =((EF)/(AF))×Cosα+((AE)/(AF))×Sinα =SinβCosα+SinαCosβ
$${EC}={EF}×{Cos}\alpha\:\:\&\:\:\:{EB}={AE}×{Sin}\alpha \\ $$$${Sin}\left(\alpha+\beta\right)=\frac{{AD}}{{AF}}=\frac{{EC}}{{AF}}+\frac{{BE}}{{AF}} \\ $$$$=\frac{{EF}}{{AF}}×{Cos}\alpha+\frac{{AE}}{{AF}}×{Sin}\alpha \\ $$$$={Sin}\beta{Cos}\alpha+{Sin}\alpha{Cos}\beta\: \\ $$
Commented by mehdee7396 last updated on 30/Jan/25
Answered by a.lgnaoui last updated on 30/Jan/25
Soit OB:Cos(a+b)i,(Sin(a+b)j Calcul de OB/Repere(O,A,A′) ci−joint demonstration :
$$\boldsymbol{\mathrm{Soit}}\:\:\boldsymbol{\mathrm{OB}}:\boldsymbol{\mathrm{Cos}}\left(\boldsymbol{\mathrm{a}}+\boldsymbol{\mathrm{b}}\right)\mathrm{i},\left(\boldsymbol{\mathrm{Sin}}\left(\boldsymbol{\mathrm{a}}+\boldsymbol{\mathrm{b}}\right)\mathrm{j}\right. \\ $$$$\boldsymbol{\mathrm{C}}\mathrm{alcul}\:\mathrm{de}\:\mathrm{OB}/\mathrm{Repere}\left(\mathrm{O},\mathrm{A},\mathrm{A}'\right) \\ $$$$\mathrm{ci}−\mathrm{joint}\:\mathrm{demonstration}\:: \\ $$
Answered by a.lgnaoui last updated on 30/Jan/25

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