How to prove the trigonometric formula $a sinθ + b cosθ = R sin(θ + α)$? [duplicate]

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  • Why does $Asink(x+c)=asinkx+bcoskx$ imply that $A=sqrta^2+b^2$ and $tanc=-b/a$?

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I recently learnt the formula $a sinθ + b cosθ = R sin(θ + α)$ at school. How to prove it? Thanks.






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    • Why does $Asink(x+c)=asinkx+bcoskx$ imply that $A=sqrta^2+b^2$ and $tanc=-b/a$?

      2 answers



    I recently learnt the formula $a sinθ + b cosθ = R sin(θ + α)$ at school. How to prove it? Thanks.






    trigonometry







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    marked as duplicate by Leucippus, Blue trigonometry
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      • Why does $Asink(x+c)=asinkx+bcoskx$ imply that $A=sqrta^2+b^2$ and $tanc=-b/a$?

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      I recently learnt the formula $a sinθ + b cosθ = R sin(θ + α)$ at school. How to prove it? Thanks.






      trigonometry







      share|cite|improve this question
















      This question already has an answer here:



      • Why does $Asink(x+c)=asinkx+bcoskx$ imply that $A=sqrta^2+b^2$ and $tanc=-b/a$?

        2 answers



      I recently learnt the formula $a sinθ + b cosθ = R sin(θ + α)$ at school. How to prove it? Thanks.





      This question already has an answer here:



      • Why does $Asink(x+c)=asinkx+bcoskx$ imply that $A=sqrta^2+b^2$ and $tanc=-b/a$?

        2 answers






      trigonometry




      trigonometry






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      edited Aug 31 at 2:43









      raindrop

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      asked Aug 31 at 2:34









      Robin Ting

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          2 Answers
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          HINT
          Suffices to show there exist $a,b,R$ such that $$asin t + bcos t = rsin(t+z).$$



          Note that
          $$
          sin(t+z) = sin t cos z + cos t sin z
          $$
          and pick $a/r = cos z, b/r = sin z$ where $a^2+b^2 = r^2$.






          share|cite|improve this answer






















          • Probably $a,b$ are given and $R,alpha$ should be found. If $a^2+b^2=0$ we may take $R=0$ whereas $alpha$ can be anything. In the non-trivial case $R:=sqrta^2+b^2$ and $alpha$ may be determined as $z$ from the above answer.
            – Jens Schwaiger
            Aug 31 at 2:50

















          up vote
          0
          down vote













          Let $dfracba=tanalpha$ ($aneq0$) then
          $$LHS=aleft(sintheta+dfracbacosthetaright)=aleft(sintheta+tanalphacosthetaright)=dfracacosalphaleft(sinthetacosalpha+sinalphacosthetaright)=dfracacosalphasin(theta+alpha)$$






          share|cite|improve this answer



























            2 Answers
            2






            active

            oldest

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            2 Answers
            2






            active

            oldest

            votes









            active

            oldest

            votes






            active

            oldest

            votes








            up vote
            0
            down vote













            HINT
            Suffices to show there exist $a,b,R$ such that $$asin t + bcos t = rsin(t+z).$$



            Note that
            $$
            sin(t+z) = sin t cos z + cos t sin z
            $$
            and pick $a/r = cos z, b/r = sin z$ where $a^2+b^2 = r^2$.






            share|cite|improve this answer






















            • Probably $a,b$ are given and $R,alpha$ should be found. If $a^2+b^2=0$ we may take $R=0$ whereas $alpha$ can be anything. In the non-trivial case $R:=sqrta^2+b^2$ and $alpha$ may be determined as $z$ from the above answer.
              – Jens Schwaiger
              Aug 31 at 2:50














            up vote
            0
            down vote













            HINT
            Suffices to show there exist $a,b,R$ such that $$asin t + bcos t = rsin(t+z).$$



            Note that
            $$
            sin(t+z) = sin t cos z + cos t sin z
            $$
            and pick $a/r = cos z, b/r = sin z$ where $a^2+b^2 = r^2$.






            share|cite|improve this answer






















            • Probably $a,b$ are given and $R,alpha$ should be found. If $a^2+b^2=0$ we may take $R=0$ whereas $alpha$ can be anything. In the non-trivial case $R:=sqrta^2+b^2$ and $alpha$ may be determined as $z$ from the above answer.
              – Jens Schwaiger
              Aug 31 at 2:50












            up vote
            0
            down vote










            up vote
            0
            down vote









            HINT
            Suffices to show there exist $a,b,R$ such that $$asin t + bcos t = rsin(t+z).$$



            Note that
            $$
            sin(t+z) = sin t cos z + cos t sin z
            $$
            and pick $a/r = cos z, b/r = sin z$ where $a^2+b^2 = r^2$.






            share|cite|improve this answer














            HINT
            Suffices to show there exist $a,b,R$ such that $$asin t + bcos t = rsin(t+z).$$



            Note that
            $$
            sin(t+z) = sin t cos z + cos t sin z
            $$
            and pick $a/r = cos z, b/r = sin z$ where $a^2+b^2 = r^2$.







            share|cite|improve this answer














            share|cite|improve this answer



            share|cite|improve this answer








            edited Aug 31 at 2:51









            steven gregory

            16.7k22155




            16.7k22155










            answered Aug 31 at 2:41









            gt6989b

            30.7k22248




            30.7k22248











            • Probably $a,b$ are given and $R,alpha$ should be found. If $a^2+b^2=0$ we may take $R=0$ whereas $alpha$ can be anything. In the non-trivial case $R:=sqrta^2+b^2$ and $alpha$ may be determined as $z$ from the above answer.
              – Jens Schwaiger
              Aug 31 at 2:50
















            • Probably $a,b$ are given and $R,alpha$ should be found. If $a^2+b^2=0$ we may take $R=0$ whereas $alpha$ can be anything. In the non-trivial case $R:=sqrta^2+b^2$ and $alpha$ may be determined as $z$ from the above answer.
              – Jens Schwaiger
              Aug 31 at 2:50















            Probably $a,b$ are given and $R,alpha$ should be found. If $a^2+b^2=0$ we may take $R=0$ whereas $alpha$ can be anything. In the non-trivial case $R:=sqrta^2+b^2$ and $alpha$ may be determined as $z$ from the above answer.
            – Jens Schwaiger
            Aug 31 at 2:50




            Probably $a,b$ are given and $R,alpha$ should be found. If $a^2+b^2=0$ we may take $R=0$ whereas $alpha$ can be anything. In the non-trivial case $R:=sqrta^2+b^2$ and $alpha$ may be determined as $z$ from the above answer.
            – Jens Schwaiger
            Aug 31 at 2:50










            up vote
            0
            down vote













            Let $dfracba=tanalpha$ ($aneq0$) then
            $$LHS=aleft(sintheta+dfracbacosthetaright)=aleft(sintheta+tanalphacosthetaright)=dfracacosalphaleft(sinthetacosalpha+sinalphacosthetaright)=dfracacosalphasin(theta+alpha)$$






            share|cite|improve this answer
























              up vote
              0
              down vote













              Let $dfracba=tanalpha$ ($aneq0$) then
              $$LHS=aleft(sintheta+dfracbacosthetaright)=aleft(sintheta+tanalphacosthetaright)=dfracacosalphaleft(sinthetacosalpha+sinalphacosthetaright)=dfracacosalphasin(theta+alpha)$$






              share|cite|improve this answer






















                up vote
                0
                down vote










                up vote
                0
                down vote









                Let $dfracba=tanalpha$ ($aneq0$) then
                $$LHS=aleft(sintheta+dfracbacosthetaright)=aleft(sintheta+tanalphacosthetaright)=dfracacosalphaleft(sinthetacosalpha+sinalphacosthetaright)=dfracacosalphasin(theta+alpha)$$






                share|cite|improve this answer












                Let $dfracba=tanalpha$ ($aneq0$) then
                $$LHS=aleft(sintheta+dfracbacosthetaright)=aleft(sintheta+tanalphacosthetaright)=dfracacosalphaleft(sinthetacosalpha+sinalphacosthetaright)=dfracacosalphasin(theta+alpha)$$







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered Aug 31 at 2:59









                Nosrati

                22k51747




                22k51747












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