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Question Number 216253 by MATHEMATICSAM last updated on 01/Feb/25

If x is a positive acute angle and sinx + sin^2 x + sin^3 x = 1 then find minimum value of cot^2 x.

$$\mathrm{If}\:{x}\:\mathrm{is}\:\mathrm{a}\:\mathrm{positive}\:\mathrm{acute}\:\mathrm{angle}\:\mathrm{and} \\ $$$$\mathrm{sin}{x}\:+\:\mathrm{sin}^{\mathrm{2}} {x}\:+\:\mathrm{sin}^{\mathrm{3}} {x}\:=\:\mathrm{1}\:\mathrm{then}\:\mathrm{find} \\ $$$$\mathrm{minimum}\:\mathrm{value}\:\mathrm{of}\:\mathrm{cot}^{\mathrm{2}} {x}. \\ $$

Commented by Frix last updated on 03/Feb/25

There′s exactly 1 solution to the given equation thus there′s exactly one value of cot^2 x sin x =((−1+(((17)/(27))−((√(33))/9))^(1/3) +(((17)/(27))+((√(33))/9))^(1/3) )/3)≈.543689 x≈32.9351° cot^2 x =(2−((2(√(33)))/9))^(1/3) +(2+((2(√(33)))/9))^(1/3) ≈2.38298

$$\mathrm{There}'\mathrm{s}\:\mathrm{exactly}\:\mathrm{1}\:\mathrm{solution}\:\mathrm{to}\:\mathrm{the}\:\mathrm{given}\:\mathrm{equation} \\ $$$$\mathrm{thus}\:\mathrm{there}'\mathrm{s}\:\mathrm{exactly}\:\mathrm{one}\:\mathrm{value}\:\mathrm{of}\:\mathrm{cot}^{\mathrm{2}} \:{x} \\ $$$$\mathrm{sin}\:{x}\:=\frac{−\mathrm{1}+\left(\frac{\mathrm{17}}{\mathrm{27}}−\frac{\sqrt{\mathrm{33}}}{\mathrm{9}}\right)^{\frac{\mathrm{1}}{\mathrm{3}}} +\left(\frac{\mathrm{17}}{\mathrm{27}}+\frac{\sqrt{\mathrm{33}}}{\mathrm{9}}\right)^{\frac{\mathrm{1}}{\mathrm{3}}} }{\mathrm{3}}\approx.\mathrm{543689} \\ $$$${x}\approx\mathrm{32}.\mathrm{9351}° \\ $$$$\mathrm{cot}^{\mathrm{2}} \:{x}\:=\left(\mathrm{2}−\frac{\mathrm{2}\sqrt{\mathrm{33}}}{\mathrm{9}}\right)^{\frac{\mathrm{1}}{\mathrm{3}}} +\left(\mathrm{2}+\frac{\mathrm{2}\sqrt{\mathrm{33}}}{\mathrm{9}}\right)^{\frac{\mathrm{1}}{\mathrm{3}}} \approx\mathrm{2}.\mathrm{38298} \\ $$

Answered by AntonCWX last updated on 03/Feb/25

let u=sin(x) u+u^2 +u^3 =1 u^3 +u^2 +u−1=0 By Lagrange′s Resolvent, ⇒z^2 +(2(1)^3 −9(1)(1)+27(−1))z+(1^2 −3(1))^3 =0 ⇒z^2 −34z−8=0 ⇒z=17∓3(√(33)) u=((−(1)+((17−3(√(33))))^(1/3) +((17+3(√(33))))^(1/3) )/3)=0.543689 sin(x)=0.543689⇒sin^2 (x)=0.295598 x=32.94° cot^2 (x)=2.38209

$${let}\:{u}={sin}\left({x}\right) \\ $$$${u}+{u}^{\mathrm{2}} +{u}^{\mathrm{3}} =\mathrm{1} \\ $$$${u}^{\mathrm{3}} +{u}^{\mathrm{2}} +{u}−\mathrm{1}=\mathrm{0} \\ $$$$ \\ $$$${By}\:{Lagrange}'{s}\:{Resolvent}, \\ $$$$\Rightarrow{z}^{\mathrm{2}} +\left(\mathrm{2}\left(\mathrm{1}\right)^{\mathrm{3}} −\mathrm{9}\left(\mathrm{1}\right)\left(\mathrm{1}\right)+\mathrm{27}\left(−\mathrm{1}\right)\right){z}+\left(\mathrm{1}^{\mathrm{2}} −\mathrm{3}\left(\mathrm{1}\right)\right)^{\mathrm{3}} =\mathrm{0} \\ $$$$\Rightarrow{z}^{\mathrm{2}} −\mathrm{34}{z}−\mathrm{8}=\mathrm{0} \\ $$$$\Rightarrow{z}=\mathrm{17}\mp\mathrm{3}\sqrt{\mathrm{33}} \\ $$$$ \\ $$$${u}=\frac{−\left(\mathrm{1}\right)+\sqrt[{\mathrm{3}}]{\mathrm{17}−\mathrm{3}\sqrt{\mathrm{33}}}+\sqrt[{\mathrm{3}}]{\mathrm{17}+\mathrm{3}\sqrt{\mathrm{33}}}}{\mathrm{3}}=\mathrm{0}.\mathrm{543689} \\ $$$$ \\ $$$${sin}\left({x}\right)=\mathrm{0}.\mathrm{543689}\Rightarrow{sin}^{\mathrm{2}} \left({x}\right)=\mathrm{0}.\mathrm{295598} \\ $$$${x}=\mathrm{32}.\mathrm{94}° \\ $$$${cot}^{\mathrm{2}} \left({x}\right)=\mathrm{2}.\mathrm{38209} \\ $$

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