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Question Number 17435 by 786786AM last updated on 06/Jul/17
Find the value of 4 sin (π/(24))  cos (π/(12))  cos(π/6).
Findthevalueof4sinπ24cosπ12cosπ6.
Answered by alex041103 last updated on 07/Jul/17
Let A=4sin(π/(24))cos(π/(12))cos(π/6).  Then we use sin2θ=2sinθcosθ :  A=2(2sin(π/(24))cos(π/(24)))cos(π/(12))cos(π/6) (1/(cos(π/(24))))  =(2sin(π/(12))cos(π/(12)))cos(π/6) (1/(cos(π/(24))))  =(1/(2cos(π/(24)))) (2sin(π/6)cos(π/6))  =((sin(π/3))/(2cos(π/(24))))  We know that cos2θ=2cos^2 θ−1  ⇒cos^2 (θ/2)=((1+cosθ)/2)  When θ∈[0,(π/2)], (θ/2)∈[0,(π/2)]⇒cos(θ/2)≧0  ⇒cos(θ/2)=(√((1+cosθ)/2))  From this we can evaluate cos(π/(24))  cos(π/(12))=(√((1+cos(π/6))/2)) = (√((1+((√3)/2))/2))  =(√((2+(√3))/4)) = ((√(2+(√3)))/2)  cos(π/(24))=(√((1+cos(π/(12)))/2)) = (√((1+((√(2+(√3)))/2))/2))  =(√((2+(√(2+(√3))))/4))=((√(2+(√(2+(√3)))))/2)  And now we solve for A:  A=(((√3)/2)/(2((√(2+(√(2+(√3)))))/2)))=((√3)/(2(√(2+(√(2+(√3)))))))  A=(1/2)(√(3/(2+(√(2+(√3))))))  A=4sin(π/(24))cos(π/(12))cos(π/6)=(1/2)(√(3/(2+(√(2+(√3))))))≈0.436749...
LetA=4sinπ24cosπ12cosπ6.Thenweusesin2θ=2sinθcosθ:A=2(2sinπ24cosπ24)cosπ12cosπ61cosπ24=(2sinπ12cosπ12)cosπ61cosπ24=12cosπ24(2sinπ6cosπ6)=sinπ32cosπ24Weknowthatcos2θ=2cos2θ1cos2θ2=1+cosθ2Whenθ[0,π2],θ2[0,π2]cosθ20cosθ2=1+cosθ2Fromthiswecanevaluatecosπ24cosπ12=1+cosπ62=1+322=2+34=2+32cosπ24=1+cosπ122=1+2+322=2+2+34=2+2+32AndnowwesolveforA:A=3222+2+32=322+2+3A=1232+2+3A=4sinπ24cosπ12cosπ6=1232+2+30.436749
Commented by mrW1 last updated on 07/Jul/17
very good sir!
verygoodsir!
Commented by alex041103 last updated on 07/Jul/17
Can you see my answer to Q:17514. I′ll  post the hole prove at some point.
CanyouseemyanswertoQ:17514.Illposttheholeproveatsomepoint.

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