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Question Number 134464 by bramlexs22 last updated on 04/Mar/21
determinant (((Solve the following Equation)),(( 81^(sin^2 x) + 81^(cos^2 x) = 30 )))
$$\:\:\:\:\:\:\:\begin{array}{|c|c|}{\mathrm{Solve}\:\mathrm{the}\:\mathrm{following}\:\mathrm{Equation}}\\{\:\:\mathrm{81}^{\mathrm{sin}\:^{\mathrm{2}} \mathrm{x}} \:+\:\mathrm{81}^{\mathrm{cos}\:^{\mathrm{2}} \mathrm{x}} \:=\:\mathrm{30}\:}\\\hline\end{array} \\ $$
Commented by harckinwunmy last updated on 04/Mar/21
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Answered by EDWIN88 last updated on 04/Mar/21
81^(sin^2 x) + 81^(1−sin^2 x) = 30 multiply both sides by 81^(sin^2 x) (81^(sin^2 x) )^2 −30×81^(sin^2 x) + 81 = 0 let 81^(sin^2 x) = u ⇒u^2 −30u + 81 = 0 (u−3)(u−27) = 0 for u=3 → determinant (((81^(sin^2 x) = 3 ⇒sin^2 x=(1/4))),((sin x=(1/2) or sin x=−(1/2))),((x= { ((π/6)),((5π/6)) :}+ 2kπ or x= { ((−π/6)),((7π/6)) :}+2kπ))) for u=27→ determinant (((81^(sin^2 x) =27⇒sin^2 x=(3/4))),((sin x = ((√3)/2) or sin x=−((√3)/2))),((x= { ((π/3)),((2π/3)) :}+2kπ or x= { ((−π/3)),((4π/3)) :}+2kπ)))
$$\:\:\:\:\:\mathrm{81}^{\mathrm{sin}\:^{\mathrm{2}} \mathrm{x}} \:+\:\mathrm{81}^{\mathrm{1}−\mathrm{sin}\:^{\mathrm{2}} \mathrm{x}} \:=\:\mathrm{30} \\ $$$$\:\mathrm{multiply}\:\mathrm{both}\:\mathrm{sides}\:\mathrm{by}\:\mathrm{81}^{\mathrm{sin}\:^{\mathrm{2}} \mathrm{x}} \\ $$$$\:\:\:\left(\mathrm{81}^{\mathrm{sin}\:^{\mathrm{2}} \mathrm{x}} \right)^{\mathrm{2}} −\mathrm{30}×\mathrm{81}^{\mathrm{sin}\:^{\mathrm{2}} \mathrm{x}} \:+\:\mathrm{81}\:=\:\mathrm{0} \\ $$$$\:\mathrm{let}\:\mathrm{81}^{\mathrm{sin}\:^{\mathrm{2}} \mathrm{x}} \:=\:\mathrm{u}\:\Rightarrow\mathrm{u}^{\mathrm{2}} −\mathrm{30u}\:+\:\mathrm{81}\:=\:\mathrm{0} \\ $$$$\left(\mathrm{u}−\mathrm{3}\right)\left(\mathrm{u}−\mathrm{27}\right)\:=\:\mathrm{0} \\ $$$$\mathrm{for}\:\mathrm{u}=\mathrm{3}\:\rightarrow\begin{array}{|c|c|c|}{\mathrm{81}^{\mathrm{sin}\:^{\mathrm{2}} \mathrm{x}} \:=\:\mathrm{3}\:\Rightarrow\mathrm{sin}\:^{\mathrm{2}} \mathrm{x}=\frac{\mathrm{1}}{\mathrm{4}}}\\{\mathrm{sin}\:\mathrm{x}=\frac{\mathrm{1}}{\mathrm{2}}\:\mathrm{or}\:\mathrm{sin}\:\mathrm{x}=−\frac{\mathrm{1}}{\mathrm{2}}}\\{\mathrm{x}=\begin{cases}{\pi/\mathrm{6}}\\{\mathrm{5}\pi/\mathrm{6}}\end{cases}+\:\mathrm{2k}\pi\:\:\mathrm{or}\:\mathrm{x}=\begin{cases}{−\pi/\mathrm{6}}\\{\mathrm{7}\pi/\mathrm{6}}\end{cases}+\mathrm{2k}\pi}\\\hline\end{array} \\ $$$$\mathrm{for}\:\mathrm{u}=\mathrm{27}\rightarrow\begin{array}{|c|c|c|}{\mathrm{81}^{\mathrm{sin}\:^{\mathrm{2}} \mathrm{x}} =\mathrm{27}\Rightarrow\mathrm{sin}\:^{\mathrm{2}} \mathrm{x}=\frac{\mathrm{3}}{\mathrm{4}}}\\{\mathrm{sin}\:\mathrm{x}\:=\:\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}\:\mathrm{or}\:\mathrm{sin}\:\mathrm{x}=−\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}}\\{\mathrm{x}=\begin{cases}{\pi/\mathrm{3}}\\{\mathrm{2}\pi/\mathrm{3}}\end{cases}+\mathrm{2k}\pi\:\mathrm{or}\:\mathrm{x}=\begin{cases}{−\pi/\mathrm{3}}\\{\mathrm{4}\pi/\mathrm{3}}\end{cases}+\mathrm{2k}\pi}\\\hline\end{array} \\ $$

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