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











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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

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















up vote
-1
down vote

favorite












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













up vote
-1
down vote

favorite









up vote
-1
down vote

favorite











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

2,859832




2,859832










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

















up vote
4
down vote



accepted










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






























    up vote
    2
    down vote













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



    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








    up vote
    4
    down vote



    accepted










    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



























      up vote
      4
      down vote



      accepted










      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

























        up vote
        4
        down vote



        accepted







        up vote
        4
        down vote



        accepted






        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

        88.8k74394




        88.8k74394






















            up vote
            2
            down vote













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



            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

















            up vote
            2
            down vote













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



            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















            up vote
            2
            down vote










            up vote
            2
            down vote









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



            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

            84919




            84919












            • 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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