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Question Number 52426 by ajfour last updated on 07/Jan/19

Commented by ajfour last updated on 07/Jan/19

Find (R/r)  if  sin θ = (5/(13)) .

FindRrifsinθ=513.

Answered by mr W last updated on 07/Jan/19

BT=(√((R+r)^2 −R^2 ))=(√(r(2R+r)))  AS=(√((R+r)^2 −r^2 ))=(√(R(R+2r)))  BP=((BT)/(sin θ))=((√(r(2R+r)))/(sin θ))  SP=((AS)/(tan θ))=((√(R(R+2r)))/(tan θ))  SP=BP+r  ((√(R(R+2r)))/(tan θ))=((√(r(2R+r)))/(sin θ))+r  let λ=(R/r)  ⇒cos θ (√(λ(λ+2)))=(√(2λ+1))+sin θ  sin θ=(5/(13))⇒cos θ=((12)/(13))  ⇒12(√(λ(λ+2)))=13(√(2λ+1))+5  ⇒λ≈2.0185

BT=(R+r)2R2=r(2R+r)AS=(R+r)2r2=R(R+2r)BP=BTsinθ=r(2R+r)sinθSP=AStanθ=R(R+2r)tanθSP=BP+rR(R+2r)tanθ=r(2R+r)sinθ+rletλ=Rrcosθλ(λ+2)=2λ+1+sinθsinθ=513cosθ=121312λ(λ+2)=132λ+1+5λ2.0185

Commented by MJS last updated on 07/Jan/19

this can be solved exactly: λ=((181)/(144))+((5(√(481)))/(144))

thiscanbesolvedexactly:λ=181144+5481144

Commented by mr W last updated on 07/Jan/19

that′s great sir! thanks!

thatsgreatsir!thanks!

Commented by ajfour last updated on 08/Jan/19

Thank you mrW Sir, Straight and Nice!  MjS Sir is very Wise.(Thanks).

ThankyoumrWSir,StraightandNice!MjSSirisveryWise.(Thanks).

Answered by mr W last updated on 07/Jan/19

(π/2)−θ=π−(sin^(−1) (R/(R+r))+cos^(−1) (r/(R+r)))  sin θ=−(r/(R+r))(√(1−((R/(R+r)))^2 ))+(R/(R+r))(√(1−((r/(R+r)))^2 ))  sin θ=((R(√(R(R+2r)))−r(√(r(2R+r))))/((R+r)^2 ))  ⇒sin θ=((λ(√(λ(λ+2)))−(√(2λ+1)))/((1+λ)^2 ))=(5/(13))  ⇒λ≈2.0185

π2θ=π(sin1RR+r+cos1rR+r)sinθ=rR+r1(RR+r)2+RR+r1(rR+r)2sinθ=RR(R+2r)rr(2R+r)(R+r)2sinθ=λλ(λ+2)2λ+1(1+λ)2=513λ2.0185

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