Simplify trigonometric expression using trigonometric identities












3












$begingroup$


I have the trigonometric expression: $$2sin x +2sin left(frac{pi} {3} -xright) $$
and it should simplified in: $$sin x + sqrt 3 cos x$$ but I do not know what formulas to apply. Could you tell me how to simplify it?






trigonometry







share|cite|improve this question











$endgroup$












  • $begingroup$
    mathworld.wolfram.com/ProsthaphaeresisFormulas.html
    $endgroup$
    – lab bhattacharjee
    Apr 1 at 15:27










  • $begingroup$
    Just between us: IMO the two expressions are of the same complexity.
    $endgroup$
    – Yves Daoust
    Apr 1 at 15:30










  • $begingroup$
    = 2 cos(x-π/6).
    $endgroup$
    – Anton Sherwood
    Apr 1 at 22:26
















3












$begingroup$


I have the trigonometric expression: $$2sin x +2sin left(frac{pi} {3} -xright) $$
and it should simplified in: $$sin x + sqrt 3 cos x$$ but I do not know what formulas to apply. Could you tell me how to simplify it?






trigonometry







share|cite|improve this question











$endgroup$












  • $begingroup$
    mathworld.wolfram.com/ProsthaphaeresisFormulas.html
    $endgroup$
    – lab bhattacharjee
    Apr 1 at 15:27










  • $begingroup$
    Just between us: IMO the two expressions are of the same complexity.
    $endgroup$
    – Yves Daoust
    Apr 1 at 15:30










  • $begingroup$
    = 2 cos(x-π/6).
    $endgroup$
    – Anton Sherwood
    Apr 1 at 22:26














3












3








3





$begingroup$


I have the trigonometric expression: $$2sin x +2sin left(frac{pi} {3} -xright) $$
and it should simplified in: $$sin x + sqrt 3 cos x$$ but I do not know what formulas to apply. Could you tell me how to simplify it?






trigonometry







share|cite|improve this question











$endgroup$




I have the trigonometric expression: $$2sin x +2sin left(frac{pi} {3} -xright) $$
and it should simplified in: $$sin x + sqrt 3 cos x$$ but I do not know what formulas to apply. Could you tell me how to simplify it?






trigonometry




trigonometry






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Apr 1 at 15:55









MarianD

2,2611618




2,2611618










asked Apr 1 at 15:19









zaz9999zaz9999

162




162












  • $begingroup$
    mathworld.wolfram.com/ProsthaphaeresisFormulas.html
    $endgroup$
    – lab bhattacharjee
    Apr 1 at 15:27










  • $begingroup$
    Just between us: IMO the two expressions are of the same complexity.
    $endgroup$
    – Yves Daoust
    Apr 1 at 15:30










  • $begingroup$
    = 2 cos(x-π/6).
    $endgroup$
    – Anton Sherwood
    Apr 1 at 22:26


















  • $begingroup$
    mathworld.wolfram.com/ProsthaphaeresisFormulas.html
    $endgroup$
    – lab bhattacharjee
    Apr 1 at 15:27










  • $begingroup$
    Just between us: IMO the two expressions are of the same complexity.
    $endgroup$
    – Yves Daoust
    Apr 1 at 15:30










  • $begingroup$
    = 2 cos(x-π/6).
    $endgroup$
    – Anton Sherwood
    Apr 1 at 22:26
















$begingroup$
mathworld.wolfram.com/ProsthaphaeresisFormulas.html
$endgroup$
– lab bhattacharjee
Apr 1 at 15:27




$begingroup$
mathworld.wolfram.com/ProsthaphaeresisFormulas.html
$endgroup$
– lab bhattacharjee
Apr 1 at 15:27












$begingroup$
Just between us: IMO the two expressions are of the same complexity.
$endgroup$
– Yves Daoust
Apr 1 at 15:30




$begingroup$
Just between us: IMO the two expressions are of the same complexity.
$endgroup$
– Yves Daoust
Apr 1 at 15:30












$begingroup$
= 2 cos(x-π/6).
$endgroup$
– Anton Sherwood
Apr 1 at 22:26




$begingroup$
= 2 cos(x-π/6).
$endgroup$
– Anton Sherwood
Apr 1 at 22:26










3 Answers
3






active

oldest

votes


















7












$begingroup$

Using the formula for $sin (alpha - beta)$ you obtain



begin{align}
2&sin x +2sin left(frac{pi} {3} -xright)\
= 2&sin x +2left[sin left(frac{pi} {3}right) cos x - cosleft(frac{pi} {3}right) sin xright]\
= 2&sin x + 2left[{sqrt 3 over 2} cos x - frac 1 2 sin xright]\[1ex]
= 2&sin x + sqrt 3 cos x - sin x\[1em] = , &color{red}{sin x + sqrt 3 cos x}
end{align}






share|cite|improve this answer











$endgroup$





















    4












    $begingroup$

    Use that $$sin(x-y)=sin(x)cos(y)-sin(y)cos(x)$$






    share|cite|improve this answer









    $endgroup$





















      1












      $begingroup$

      Hint: $sin(frac{pi}{3}-x)=sin frac{pi}{3}cos x-cos frac{pi}{3}sin x$






      share|cite|improve this answer









      $endgroup$














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        3 Answers
        3






        active

        oldest

        votes








        3 Answers
        3






        active

        oldest

        votes









        active

        oldest

        votes






        active

        oldest

        votes









        7












        $begingroup$

        Using the formula for $sin (alpha - beta)$ you obtain



        begin{align}
        2&sin x +2sin left(frac{pi} {3} -xright)\
        = 2&sin x +2left[sin left(frac{pi} {3}right) cos x - cosleft(frac{pi} {3}right) sin xright]\
        = 2&sin x + 2left[{sqrt 3 over 2} cos x - frac 1 2 sin xright]\[1ex]
        = 2&sin x + sqrt 3 cos x - sin x\[1em] = , &color{red}{sin x + sqrt 3 cos x}
        end{align}






        share|cite|improve this answer











        $endgroup$


















          7












          $begingroup$

          Using the formula for $sin (alpha - beta)$ you obtain



          begin{align}
          2&sin x +2sin left(frac{pi} {3} -xright)\
          = 2&sin x +2left[sin left(frac{pi} {3}right) cos x - cosleft(frac{pi} {3}right) sin xright]\
          = 2&sin x + 2left[{sqrt 3 over 2} cos x - frac 1 2 sin xright]\[1ex]
          = 2&sin x + sqrt 3 cos x - sin x\[1em] = , &color{red}{sin x + sqrt 3 cos x}
          end{align}






          share|cite|improve this answer











          $endgroup$
















            7












            7








            7





            $begingroup$

            Using the formula for $sin (alpha - beta)$ you obtain



            begin{align}
            2&sin x +2sin left(frac{pi} {3} -xright)\
            = 2&sin x +2left[sin left(frac{pi} {3}right) cos x - cosleft(frac{pi} {3}right) sin xright]\
            = 2&sin x + 2left[{sqrt 3 over 2} cos x - frac 1 2 sin xright]\[1ex]
            = 2&sin x + sqrt 3 cos x - sin x\[1em] = , &color{red}{sin x + sqrt 3 cos x}
            end{align}






            share|cite|improve this answer











            $endgroup$



            Using the formula for $sin (alpha - beta)$ you obtain



            begin{align}
            2&sin x +2sin left(frac{pi} {3} -xright)\
            = 2&sin x +2left[sin left(frac{pi} {3}right) cos x - cosleft(frac{pi} {3}right) sin xright]\
            = 2&sin x + 2left[{sqrt 3 over 2} cos x - frac 1 2 sin xright]\[1ex]
            = 2&sin x + sqrt 3 cos x - sin x\[1em] = , &color{red}{sin x + sqrt 3 cos x}
            end{align}







            share|cite|improve this answer














            share|cite|improve this answer



            share|cite|improve this answer








            edited Apr 1 at 15:59

























            answered Apr 1 at 15:44









            MarianDMarianD

            2,2611618




            2,2611618























                4












                $begingroup$

                Use that $$sin(x-y)=sin(x)cos(y)-sin(y)cos(x)$$






                share|cite|improve this answer









                $endgroup$


















                  4












                  $begingroup$

                  Use that $$sin(x-y)=sin(x)cos(y)-sin(y)cos(x)$$






                  share|cite|improve this answer









                  $endgroup$
















                    4












                    4








                    4





                    $begingroup$

                    Use that $$sin(x-y)=sin(x)cos(y)-sin(y)cos(x)$$






                    share|cite|improve this answer









                    $endgroup$



                    Use that $$sin(x-y)=sin(x)cos(y)-sin(y)cos(x)$$







                    share|cite|improve this answer












                    share|cite|improve this answer



                    share|cite|improve this answer










                    answered Apr 1 at 15:26









                    Dr. Sonnhard GraubnerDr. Sonnhard Graubner

                    79k42867




                    79k42867























                        1












                        $begingroup$

                        Hint: $sin(frac{pi}{3}-x)=sin frac{pi}{3}cos x-cos frac{pi}{3}sin x$






                        share|cite|improve this answer









                        $endgroup$


















                          1












                          $begingroup$

                          Hint: $sin(frac{pi}{3}-x)=sin frac{pi}{3}cos x-cos frac{pi}{3}sin x$






                          share|cite|improve this answer









                          $endgroup$
















                            1












                            1








                            1





                            $begingroup$

                            Hint: $sin(frac{pi}{3}-x)=sin frac{pi}{3}cos x-cos frac{pi}{3}sin x$






                            share|cite|improve this answer









                            $endgroup$



                            Hint: $sin(frac{pi}{3}-x)=sin frac{pi}{3}cos x-cos frac{pi}{3}sin x$







                            share|cite|improve this answer












                            share|cite|improve this answer



                            share|cite|improve this answer










                            answered Apr 1 at 15:25









                            VasyaVasya

                            4,4671618




                            4,4671618






























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