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Question-188912




Question Number 188912 by Mingma last updated on 08/Mar/23
Answered by a.lgnaoui last updated on 09/Mar/23
posons  (θ/3)=x  sin (((π−θ)/3))=sin ((π/3)−x)=(1/2)((√3)cos x−sin x)  sin (((π+θ)/3))=sin ((π/3)+x)=(1/2)((√3)cos x+sin x)      sin^3 x −(3/4)sin x+((1+(√5))/(16))=0  •sin x=−0,9781476..         (1)  •sin x=+0,3090169..          (2)  •sin x=+0,6691306..           (3)    (1): x=−78°      ⇒   θ=−234  (2): x=      18°       ⇒   θ=       54  (3): x=       42       ⇒    θ=     126     for   0<θ<π    Solutions={54;126}
$${posons}\:\:\frac{\theta}{\mathrm{3}}={x} \\ $$$$\mathrm{sin}\:\left(\frac{\pi−\theta}{\mathrm{3}}\right)=\mathrm{sin}\:\left(\frac{\pi}{\mathrm{3}}−{x}\right)=\frac{\mathrm{1}}{\mathrm{2}}\left(\sqrt{\mathrm{3}}\mathrm{cos}\:{x}−\mathrm{sin}\:{x}\right) \\ $$$$\mathrm{sin}\:\left(\frac{\pi+\theta}{\mathrm{3}}\right)=\mathrm{sin}\:\left(\frac{\pi}{\mathrm{3}}+{x}\right)=\frac{\mathrm{1}}{\mathrm{2}}\left(\sqrt{\mathrm{3}}\mathrm{cos}\:{x}+\mathrm{sin}\:{x}\right) \\ $$$$\:\: \\ $$$$\mathrm{sin}^{\mathrm{3}} {x}\:−\frac{\mathrm{3}}{\mathrm{4}}\mathrm{sin}\:{x}+\frac{\mathrm{1}+\sqrt{\mathrm{5}}}{\mathrm{16}}=\mathrm{0} \\ $$$$\bullet\mathrm{sin}\:{x}=−\mathrm{0},\mathrm{9781476}..\:\:\:\:\:\:\:\:\:\left(\mathrm{1}\right) \\ $$$$\bullet\mathrm{sin}\:{x}=+\mathrm{0},\mathrm{3090169}..\:\:\:\:\:\:\:\:\:\:\left(\mathrm{2}\right) \\ $$$$\bullet\mathrm{sin}\:{x}=+\mathrm{0},\mathrm{6691306}..\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{3}\right) \\ $$$$ \\ $$$$\left(\mathrm{1}\right):\:{x}=−\mathrm{78}°\:\:\:\:\:\:\Rightarrow\:\:\:\theta=−\mathrm{234} \\ $$$$\left(\mathrm{2}\right):\:{x}=\:\:\:\:\:\:\mathrm{18}°\:\:\:\:\:\:\:\Rightarrow\:\:\:\theta=\:\:\:\:\:\:\:\mathrm{54} \\ $$$$\left(\mathrm{3}\right):\:{x}=\:\:\:\:\:\:\:\mathrm{42}\:\:\:\:\:\:\:\Rightarrow\:\:\:\:\theta=\:\:\:\:\:\mathrm{126} \\ $$$$\:\:\:{for}\:\:\:\mathrm{0}<\theta<\pi \\ $$$$\:\:\boldsymbol{{Solutions}}=\left\{\mathrm{54};\mathrm{126}\right\}\:\: \\ $$
Commented by a.lgnaoui last updated on 08/Mar/23
Commented by Mingma last updated on 09/Mar/23
Excellent!
Answered by mr W last updated on 09/Mar/23
recall   sin α sin β=(1/2)[cos (α−β)−cos (α+β)]  sin 3x=sin x(3−4 sin^2  x)  sin 54°=sin ((3π)/(10))=((1+(√5))/4)    let x=(θ/3)  sin ((π/3)−x) sin ((π/3)+x)  =(1/2)[cos (−2x)−cos ((2π)/3)]  =(1/2)[cos (2x)+(1/2)]  =(1/4)(3−4 sin^2  x)  eqn. becomes to  (1/4)(3−4 sin^2  x)sin x=((1+(√5))/(16))  (3−4 sin^2  x)sin x=((1+(√5))/4)  sin 3x=((1+(√5))/4)  sin θ=((1+(√5))/4)  ⇒θ=nπ+(−1)^n  sin^(−1) ((1+(√5))/4)  ⇒θ=nπ+(((−1)^n 3π)/(10))   within 0<θ<π  ⇒θ=((3π)/(10)) or ((7π)/(10)), i.e. 54° or 126°
$${recall}\: \\ $$$$\mathrm{sin}\:\alpha\:\mathrm{sin}\:\beta=\frac{\mathrm{1}}{\mathrm{2}}\left[\mathrm{cos}\:\left(\alpha−\beta\right)−\mathrm{cos}\:\left(\alpha+\beta\right)\right] \\ $$$$\mathrm{sin}\:\mathrm{3}{x}=\mathrm{sin}\:{x}\left(\mathrm{3}−\mathrm{4}\:\mathrm{sin}^{\mathrm{2}} \:{x}\right) \\ $$$$\mathrm{sin}\:\mathrm{54}°=\mathrm{sin}\:\frac{\mathrm{3}\pi}{\mathrm{10}}=\frac{\mathrm{1}+\sqrt{\mathrm{5}}}{\mathrm{4}} \\ $$$$ \\ $$$${let}\:{x}=\frac{\theta}{\mathrm{3}} \\ $$$$\mathrm{sin}\:\left(\frac{\pi}{\mathrm{3}}−{x}\right)\:\mathrm{sin}\:\left(\frac{\pi}{\mathrm{3}}+{x}\right) \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\left[\mathrm{cos}\:\left(−\mathrm{2}{x}\right)−\mathrm{cos}\:\frac{\mathrm{2}\pi}{\mathrm{3}}\right] \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\left[\mathrm{cos}\:\left(\mathrm{2}{x}\right)+\frac{\mathrm{1}}{\mathrm{2}}\right] \\ $$$$=\frac{\mathrm{1}}{\mathrm{4}}\left(\mathrm{3}−\mathrm{4}\:\mathrm{sin}^{\mathrm{2}} \:{x}\right) \\ $$$${eqn}.\:{becomes}\:{to} \\ $$$$\frac{\mathrm{1}}{\mathrm{4}}\left(\mathrm{3}−\mathrm{4}\:\mathrm{sin}^{\mathrm{2}} \:{x}\right)\mathrm{sin}\:{x}=\frac{\mathrm{1}+\sqrt{\mathrm{5}}}{\mathrm{16}} \\ $$$$\left(\mathrm{3}−\mathrm{4}\:\mathrm{sin}^{\mathrm{2}} \:{x}\right)\mathrm{sin}\:{x}=\frac{\mathrm{1}+\sqrt{\mathrm{5}}}{\mathrm{4}} \\ $$$$\mathrm{sin}\:\mathrm{3}{x}=\frac{\mathrm{1}+\sqrt{\mathrm{5}}}{\mathrm{4}} \\ $$$$\mathrm{sin}\:\theta=\frac{\mathrm{1}+\sqrt{\mathrm{5}}}{\mathrm{4}} \\ $$$$\Rightarrow\theta={n}\pi+\left(−\mathrm{1}\right)^{{n}} \:\mathrm{sin}^{−\mathrm{1}} \frac{\mathrm{1}+\sqrt{\mathrm{5}}}{\mathrm{4}} \\ $$$$\Rightarrow\theta={n}\pi+\frac{\left(−\mathrm{1}\right)^{{n}} \mathrm{3}\pi}{\mathrm{10}}\: \\ $$$${within}\:\mathrm{0}<\theta<\pi \\ $$$$\Rightarrow\theta=\frac{\mathrm{3}\pi}{\mathrm{10}}\:{or}\:\frac{\mathrm{7}\pi}{\mathrm{10}},\:{i}.{e}.\:\mathrm{54}°\:{or}\:\mathrm{126}° \\ $$
Commented by manxsol last updated on 09/Mar/23
Clear job. Perfect!
$${Clear}\:{job}.\:{Perfect}! \\ $$
Answered by Frix last updated on 09/Mar/23
Use trigonometric formulas to get  sin ((π−θ)/3) sin (θ/3) sin ((π+θ)/3) =((sin x)/4)  sin x=((1+(√5))/4) ∧ 0<x<π ⇒  x=((3π)/(10))∨x=((7π)/(10))
$$\mathrm{Use}\:\mathrm{trigonometric}\:\mathrm{formulas}\:\mathrm{to}\:\mathrm{get} \\ $$$$\mathrm{sin}\:\frac{\pi−\theta}{\mathrm{3}}\:\mathrm{sin}\:\frac{\theta}{\mathrm{3}}\:\mathrm{sin}\:\frac{\pi+\theta}{\mathrm{3}}\:=\frac{\mathrm{sin}\:{x}}{\mathrm{4}} \\ $$$$\mathrm{sin}\:{x}=\frac{\mathrm{1}+\sqrt{\mathrm{5}}}{\mathrm{4}}\:\wedge\:\mathrm{0}<{x}<\pi\:\Rightarrow \\ $$$${x}=\frac{\mathrm{3}\pi}{\mathrm{10}}\vee{x}=\frac{\mathrm{7}\pi}{\mathrm{10}} \\ $$
Answered by mnjuly1970 last updated on 09/Mar/23
   sin(x ).sin((π/3) −x ).sin((π/3)+x )=^(easy)  ((sin(3x))/4)     x → (x/3)        sin((x/3)).sin((π/3) −(x/3) ).sin((π/3)+(x/3))=(1/4)sin(x)       x= (π/(10))
$$ \\ $$$$\:{sin}\left({x}\:\right).{sin}\left(\frac{\pi}{\mathrm{3}}\:−{x}\:\right).{sin}\left(\frac{\pi}{\mathrm{3}}+{x}\:\right)\overset{{easy}} {=}\:\frac{{sin}\left(\mathrm{3}{x}\right)}{\mathrm{4}} \\ $$$$\:\:\:{x}\:\rightarrow\:\frac{{x}}{\mathrm{3}} \\ $$$$\:\:\:\:\:\:{sin}\left(\frac{{x}}{\mathrm{3}}\right).{sin}\left(\frac{\pi}{\mathrm{3}}\:−\frac{{x}}{\mathrm{3}}\:\right).{sin}\left(\frac{\pi}{\mathrm{3}}+\frac{{x}}{\mathrm{3}}\right)=\frac{\mathrm{1}}{\mathrm{4}}{sin}\left({x}\right) \\ $$$$\:\:\:\:\:{x}=\:\frac{\pi}{\mathrm{10}} \\ $$

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