Given $sin(x)$, find $sin(frac{x}{2}) cos(frac{5x}{2})$ [closed]












-1














Knowing that
$pi < 2x < 2pi$
and



$$sin(x) = frac{4}{5},$$



find



$$sinleft(frac{x}{2}right) cosleft(frac{5x}{2}right) = ?$$










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closed as off-topic by Nosrati, Davide Giraudo, amWhy, Leucippus, Cesareo Nov 23 at 1:10


This question appears to be off-topic. The users who voted to close gave this specific reason:


  • "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Nosrati, Davide Giraudo, amWhy, Leucippus, Cesareo

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  • 1




    mathworld.wolfram.com/WernerFormulas.html
    – lab bhattacharjee
    Nov 22 at 14:17
















-1














Knowing that
$pi < 2x < 2pi$
and



$$sin(x) = frac{4}{5},$$



find



$$sinleft(frac{x}{2}right) cosleft(frac{5x}{2}right) = ?$$










share|cite|improve this question















closed as off-topic by Nosrati, Davide Giraudo, amWhy, Leucippus, Cesareo Nov 23 at 1:10


This question appears to be off-topic. The users who voted to close gave this specific reason:


  • "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Nosrati, Davide Giraudo, amWhy, Leucippus, Cesareo

If this question can be reworded to fit the rules in the help center, please edit the question.









  • 1




    mathworld.wolfram.com/WernerFormulas.html
    – lab bhattacharjee
    Nov 22 at 14:17














-1












-1








-1







Knowing that
$pi < 2x < 2pi$
and



$$sin(x) = frac{4}{5},$$



find



$$sinleft(frac{x}{2}right) cosleft(frac{5x}{2}right) = ?$$










share|cite|improve this question















Knowing that
$pi < 2x < 2pi$
and



$$sin(x) = frac{4}{5},$$



find



$$sinleft(frac{x}{2}right) cosleft(frac{5x}{2}right) = ?$$







trigonometry transformation






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share|cite|improve this question













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edited Nov 22 at 14:13









Tianlalu

3,02021038




3,02021038










asked Nov 22 at 14:07









critical_mass

81




81




closed as off-topic by Nosrati, Davide Giraudo, amWhy, Leucippus, Cesareo Nov 23 at 1:10


This question appears to be off-topic. The users who voted to close gave this specific reason:


  • "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Nosrati, Davide Giraudo, amWhy, Leucippus, Cesareo

If this question can be reworded to fit the rules in the help center, please edit the question.




closed as off-topic by Nosrati, Davide Giraudo, amWhy, Leucippus, Cesareo Nov 23 at 1:10


This question appears to be off-topic. The users who voted to close gave this specific reason:


  • "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Nosrati, Davide Giraudo, amWhy, Leucippus, Cesareo

If this question can be reworded to fit the rules in the help center, please edit the question.








  • 1




    mathworld.wolfram.com/WernerFormulas.html
    – lab bhattacharjee
    Nov 22 at 14:17














  • 1




    mathworld.wolfram.com/WernerFormulas.html
    – lab bhattacharjee
    Nov 22 at 14:17








1




1




mathworld.wolfram.com/WernerFormulas.html
– lab bhattacharjee
Nov 22 at 14:17




mathworld.wolfram.com/WernerFormulas.html
– lab bhattacharjee
Nov 22 at 14:17










2 Answers
2






active

oldest

votes


















4














HINT



Use that



$$sin theta cos varphi = frac12{{sin(theta + varphi) + frac12 sin(theta - varphi)} }$$



and




  • $sin (2theta) =2sintheta costheta$


  • $sin (3theta) =3sintheta - 4sin^3theta$



moreover from the given




  • $cos x=-sqrt{1-sin^2 x}$






share|cite|improve this answer































    2














    $sin(x/2)cos(5x/2)\
    =sin(x/2)cos(2x+x/2)\=sin(x/2){cos(2x)cos(x/2)-sin(2x)sin(x/2)}\=cos(2x)cos(x/2)sin(x/2)-sin(2x)sin^2(x/2)\=1/2(1-2sin^2(x))sin(x)-sin(x)cos(x)(1-cos(x))$



    As $pi/2<x<pi$, $sin(x)=4/5$ implies $cos(x)=-3/5$. So the final answer is $sin(x/2)cos(5x/2)=82/125.$






    share|cite|improve this answer





















    • Welcome to to MathSE. Type $sin x$, $cos x$, $tan x$, $csc x$, $sec x$, and $cot x$ to produce $sin x$, $cos x$, $tan x$, $csc x$, $sec x$, and $cot x$, respectively.
      – N. F. Taussig
      Nov 22 at 14:34




















    2 Answers
    2






    active

    oldest

    votes








    2 Answers
    2






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes









    4














    HINT



    Use that



    $$sin theta cos varphi = frac12{{sin(theta + varphi) + frac12 sin(theta - varphi)} }$$



    and




    • $sin (2theta) =2sintheta costheta$


    • $sin (3theta) =3sintheta - 4sin^3theta$



    moreover from the given




    • $cos x=-sqrt{1-sin^2 x}$






    share|cite|improve this answer




























      4














      HINT



      Use that



      $$sin theta cos varphi = frac12{{sin(theta + varphi) + frac12 sin(theta - varphi)} }$$



      and




      • $sin (2theta) =2sintheta costheta$


      • $sin (3theta) =3sintheta - 4sin^3theta$



      moreover from the given




      • $cos x=-sqrt{1-sin^2 x}$






      share|cite|improve this answer


























        4












        4








        4






        HINT



        Use that



        $$sin theta cos varphi = frac12{{sin(theta + varphi) + frac12 sin(theta - varphi)} }$$



        and




        • $sin (2theta) =2sintheta costheta$


        • $sin (3theta) =3sintheta - 4sin^3theta$



        moreover from the given




        • $cos x=-sqrt{1-sin^2 x}$






        share|cite|improve this answer














        HINT



        Use that



        $$sin theta cos varphi = frac12{{sin(theta + varphi) + frac12 sin(theta - varphi)} }$$



        and




        • $sin (2theta) =2sintheta costheta$


        • $sin (3theta) =3sintheta - 4sin^3theta$



        moreover from the given




        • $cos x=-sqrt{1-sin^2 x}$







        share|cite|improve this answer














        share|cite|improve this answer



        share|cite|improve this answer








        edited Nov 22 at 14:13

























        answered Nov 22 at 14:10









        gimusi

        1




        1























            2














            $sin(x/2)cos(5x/2)\
            =sin(x/2)cos(2x+x/2)\=sin(x/2){cos(2x)cos(x/2)-sin(2x)sin(x/2)}\=cos(2x)cos(x/2)sin(x/2)-sin(2x)sin^2(x/2)\=1/2(1-2sin^2(x))sin(x)-sin(x)cos(x)(1-cos(x))$



            As $pi/2<x<pi$, $sin(x)=4/5$ implies $cos(x)=-3/5$. So the final answer is $sin(x/2)cos(5x/2)=82/125.$






            share|cite|improve this answer





















            • Welcome to to MathSE. Type $sin x$, $cos x$, $tan x$, $csc x$, $sec x$, and $cot x$ to produce $sin x$, $cos x$, $tan x$, $csc x$, $sec x$, and $cot x$, respectively.
              – N. F. Taussig
              Nov 22 at 14:34


















            2














            $sin(x/2)cos(5x/2)\
            =sin(x/2)cos(2x+x/2)\=sin(x/2){cos(2x)cos(x/2)-sin(2x)sin(x/2)}\=cos(2x)cos(x/2)sin(x/2)-sin(2x)sin^2(x/2)\=1/2(1-2sin^2(x))sin(x)-sin(x)cos(x)(1-cos(x))$



            As $pi/2<x<pi$, $sin(x)=4/5$ implies $cos(x)=-3/5$. So the final answer is $sin(x/2)cos(5x/2)=82/125.$






            share|cite|improve this answer





















            • Welcome to to MathSE. Type $sin x$, $cos x$, $tan x$, $csc x$, $sec x$, and $cot x$ to produce $sin x$, $cos x$, $tan x$, $csc x$, $sec x$, and $cot x$, respectively.
              – N. F. Taussig
              Nov 22 at 14:34
















            2












            2








            2






            $sin(x/2)cos(5x/2)\
            =sin(x/2)cos(2x+x/2)\=sin(x/2){cos(2x)cos(x/2)-sin(2x)sin(x/2)}\=cos(2x)cos(x/2)sin(x/2)-sin(2x)sin^2(x/2)\=1/2(1-2sin^2(x))sin(x)-sin(x)cos(x)(1-cos(x))$



            As $pi/2<x<pi$, $sin(x)=4/5$ implies $cos(x)=-3/5$. So the final answer is $sin(x/2)cos(5x/2)=82/125.$






            share|cite|improve this answer












            $sin(x/2)cos(5x/2)\
            =sin(x/2)cos(2x+x/2)\=sin(x/2){cos(2x)cos(x/2)-sin(2x)sin(x/2)}\=cos(2x)cos(x/2)sin(x/2)-sin(2x)sin^2(x/2)\=1/2(1-2sin^2(x))sin(x)-sin(x)cos(x)(1-cos(x))$



            As $pi/2<x<pi$, $sin(x)=4/5$ implies $cos(x)=-3/5$. So the final answer is $sin(x/2)cos(5x/2)=82/125.$







            share|cite|improve this answer












            share|cite|improve this answer



            share|cite|improve this answer










            answered Nov 22 at 14:23









            John_Wick

            1,314111




            1,314111












            • Welcome to to MathSE. Type $sin x$, $cos x$, $tan x$, $csc x$, $sec x$, and $cot x$ to produce $sin x$, $cos x$, $tan x$, $csc x$, $sec x$, and $cot x$, respectively.
              – N. F. Taussig
              Nov 22 at 14:34




















            • Welcome to to MathSE. Type $sin x$, $cos x$, $tan x$, $csc x$, $sec x$, and $cot x$ to produce $sin x$, $cos x$, $tan x$, $csc x$, $sec x$, and $cot x$, respectively.
              – N. F. Taussig
              Nov 22 at 14:34


















            Welcome to to MathSE. Type $sin x$, $cos x$, $tan x$, $csc x$, $sec x$, and $cot x$ to produce $sin x$, $cos x$, $tan x$, $csc x$, $sec x$, and $cot x$, respectively.
            – N. F. Taussig
            Nov 22 at 14:34






            Welcome to to MathSE. Type $sin x$, $cos x$, $tan x$, $csc x$, $sec x$, and $cot x$ to produce $sin x$, $cos x$, $tan x$, $csc x$, $sec x$, and $cot x$, respectively.
            – N. F. Taussig
            Nov 22 at 14:34





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