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Question Number 159005 by Tawa11 last updated on 11/Nov/21

Answered by ajfour last updated on 12/Nov/21

s=2rsin α p=rsin 2α cos β=((35/2)/r) =(7/8); sin β=((√(15))/8) ((sin (90°−α))/(27))=((sin (3α+β))/p) 20sin 2αcos α=27sin (3α+β) 10sin α+37sin 3α=27sin (3α+β)+27sin 3α =27cos β{3sin α−4sin^3 α +((√(15))/7)(4cos^3 α−3cos α)} say tan α=m 40=27×(7/8){3(1+m^2 )−4m^3 +((√(15))/7)[4−3(1+m^2 )]} ...we need a straight cubic formula...still trying!

$${s}=\mathrm{2}{r}\mathrm{sin}\:\alpha \\ $$$${p}={r}\mathrm{sin}\:\mathrm{2}\alpha \\ $$$$\mathrm{cos}\:\beta=\frac{\mathrm{35}/\mathrm{2}}{{r}}\:=\frac{\mathrm{7}}{\mathrm{8}};\:\:\mathrm{sin}\:\beta=\frac{\sqrt{\mathrm{15}}}{\mathrm{8}} \\ $$$$\frac{\mathrm{sin}\:\left(\mathrm{90}°−\alpha\right)}{\mathrm{27}}=\frac{\mathrm{sin}\:\left(\mathrm{3}\alpha+\beta\right)}{{p}} \\ $$$$\mathrm{20sin}\:\mathrm{2}\alpha\mathrm{cos}\:\alpha=\mathrm{27sin}\:\left(\mathrm{3}\alpha+\beta\right) \\ $$$$\mathrm{10sin}\:\alpha+\mathrm{37sin}\:\mathrm{3}\alpha=\mathrm{27sin}\:\left(\mathrm{3}\alpha+\beta\right)+\mathrm{27sin}\:\mathrm{3}\alpha \\ $$$$\:\:\:=\mathrm{27cos}\:\beta\left\{\mathrm{3sin}\:\alpha−\mathrm{4sin}\:^{\mathrm{3}} \alpha\right. \\ $$$$\left.\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:+\frac{\sqrt{\mathrm{15}}}{\mathrm{7}}\left(\mathrm{4cos}\:^{\mathrm{3}} \alpha−\mathrm{3cos}\:\alpha\right)\right\} \\ $$$${say}\:\:\mathrm{tan}\:\alpha={m} \\ $$$$\mathrm{40}=\mathrm{27}×\frac{\mathrm{7}}{\mathrm{8}}\left\{\mathrm{3}\left(\mathrm{1}+{m}^{\mathrm{2}} \right)−\mathrm{4}{m}^{\mathrm{3}} \right. \\ $$$$\left.\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:+\frac{\sqrt{\mathrm{15}}}{\mathrm{7}}\left[\mathrm{4}−\mathrm{3}\left(\mathrm{1}+{m}^{\mathrm{2}} \right)\right]\right\} \\ $$$$...{we}\:{need}\:{a}\:{straight}\:{cubic}\: \\ $$$${formula}...{still}\:{trying}! \\ $$

Commented by Tawa11 last updated on 12/Nov/21

God bless you sir.

$$\mathrm{God}\:\mathrm{bless}\:\mathrm{you}\:\mathrm{sir}. \\ $$

Answered by mr W last updated on 11/Nov/21

Commented by mr W last updated on 12/Nov/21

s_1 s_2 =27×8 s_1 +s_2 =s s_2 +((216)/s_2 )=s s_2 ^2 −ss_2 =−216 s_2 =((s+(√(s^2 −864)))/2) s=2×20 cos (θ/2) ⇒cos (θ/2)=(s/(40)) 27^2 =s^2 +s_2 ^2 −2ss_2 cos θ 27^2 =s^2 +s_2 ^2 −ss_2 +ss_2 (2cos^2 (θ/2)−1) 27^2 +216=s^2 +ss_2 (2cos^2 (θ/2)−1) s^2 +((s^2 +s(√(s^2 −864)))/2)((s^2 /(800))−1)=945 ⇒s≈29.8841

$${s}_{\mathrm{1}} {s}_{\mathrm{2}} =\mathrm{27}×\mathrm{8} \\ $$$${s}_{\mathrm{1}} +{s}_{\mathrm{2}} ={s} \\ $$$${s}_{\mathrm{2}} +\frac{\mathrm{216}}{{s}_{\mathrm{2}} }={s} \\ $$$${s}_{\mathrm{2}} ^{\mathrm{2}} −{ss}_{\mathrm{2}} =−\mathrm{216} \\ $$$${s}_{\mathrm{2}} =\frac{{s}+\sqrt{{s}^{\mathrm{2}} −\mathrm{864}}}{\mathrm{2}} \\ $$$${s}=\mathrm{2}×\mathrm{20}\:\mathrm{cos}\:\frac{\theta}{\mathrm{2}} \\ $$$$\Rightarrow\mathrm{cos}\:\frac{\theta}{\mathrm{2}}=\frac{{s}}{\mathrm{40}} \\ $$$$\mathrm{27}^{\mathrm{2}} ={s}^{\mathrm{2}} +{s}_{\mathrm{2}} ^{\mathrm{2}} −\mathrm{2}{ss}_{\mathrm{2}} \mathrm{cos}\:\theta \\ $$$$\mathrm{27}^{\mathrm{2}} ={s}^{\mathrm{2}} +{s}_{\mathrm{2}} ^{\mathrm{2}} −{ss}_{\mathrm{2}} +{ss}_{\mathrm{2}} \left(\mathrm{2cos}^{\mathrm{2}} \:\frac{\theta}{\mathrm{2}}−\mathrm{1}\right) \\ $$$$\mathrm{27}^{\mathrm{2}} +\mathrm{216}={s}^{\mathrm{2}} +{ss}_{\mathrm{2}} \left(\mathrm{2cos}^{\mathrm{2}} \:\frac{\theta}{\mathrm{2}}−\mathrm{1}\right) \\ $$$${s}^{\mathrm{2}} +\frac{{s}^{\mathrm{2}} +{s}\sqrt{{s}^{\mathrm{2}} −\mathrm{864}}}{\mathrm{2}}\left(\frac{{s}^{\mathrm{2}} }{\mathrm{800}}−\mathrm{1}\right)=\mathrm{945} \\ $$$$\Rightarrow{s}\approx\mathrm{29}.\mathrm{8841} \\ $$

Commented by Tawa11 last updated on 12/Nov/21

God bless you sir.

$$\mathrm{God}\:\mathrm{bless}\:\mathrm{you}\:\mathrm{sir}. \\ $$

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