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Question Number 59389 by hovea cw last updated on 09/May/19

$$\mathrm{If}0⩽x⩽\pi \mathrm{and}{81}^{{\sin }^{2}x}+{81}^{{\cos }^{2}x}=30,$$$$\mathrm{then}x\mathrm{is}\mathrm{equal}\mathrm{to}$$

Answered by ajfour last updated on 09/May/19

$$\mathrm{t}+\frac{81}{\mathrm{t}}=30$$$$\Rightarrow {\mathrm{t}}^2-30\mathrm{t}+81=0$$$$\Rightarrow \mathrm{t}=3,27$$$${81}^{\sin {}^2\mathrm{x}}={81}^{\frac{1}{4}},{81}^{\frac{3}{4}}$$$$\Rightarrow \mathrm{x}=\frac{\pi }{6},\frac{\pi }{3},\frac{2\pi }{3},\frac{5\pi }{6}.$$$$$$

Answered by tanmay last updated on 09/May/19

$${\left(81\right)}^{{sin}^{2}x}+{\left(81\right)}^{1-{sin}^{2}x}=30$$$${\left(81\right)}^{{sin}^{2}x}+\frac{81}{{\left(81\right)}^{{sin}^{2}x}}=30$$$${\left(9\right)}^{2{sin}^{2}x}+\frac{9^2}{{\left(9\right)}^{2{sin}^{2}x}}=30$$$${(9^{{sin}^{2}x}-\frac{9}{9^{{sin}^{2}x}})}^2+2\times 9^{{sin}^{2}x}\times \frac{9}{9^{{sin}^{2}x}}=30$$$${(9^{{sin}^{2}x}-\frac{9}{9^{{sin}^{2}x}})}^2={\left(2\sqrt{3}\right)}^2$$$${(k-\frac{9}{k})}^2-{\left(2\sqrt{3}\right)}^2=0$$$$k-\frac{9}{k}+2\sqrt{3}=0$$$$k^2+2\sqrt{3}k-9=0$$$$k^2+(3\sqrt{3}-\sqrt{3})k-3\times \sqrt{3}\times \sqrt{3}=0$$$$k(k+3\sqrt{3})-\sqrt{3}(k+3\sqrt{3})=0$$$$(k+3\sqrt{3})(k-\sqrt{3})=0$$$$k+3\sqrt{3}\ne 0$$$$k=\sqrt{3}$$$$9^{{sin}^{2}x}=3^{\frac{1}{2}}$$$$3^{2{sin}^{2}x}=3^{\frac{1}{2}}$$$${sin}^{2}x=\frac{1}{4}$$$$sinx=\frac{1}{2}=sin\left(\frac{\pi }{6}\right)\to x=\frac{\pi }{6}\left(firstquadrant\right)$$$$and(\pi -\frac{\pi }{6})i,e\frac{5\pi }{6}insecondquadrant$$$$sinx\ne -\frac{1}{2}when\pi ⩾x⩾0$$$$again$$$$k-\frac{9}{k}-2\sqrt{3}=0$$$$k^2-9-2\sqrt{3}k=0$$$$k^2-2\sqrt{3}k+3-12=0$$$${(k-\sqrt{3})}^2-{\left(2\sqrt{3}\right)}^2=0$$$$(k-\sqrt{3}+2\sqrt{3})(k-\sqrt{3}-2\sqrt{3})=0$$$$k+\sqrt{3}\ne 0$$$$k=3\sqrt{3}$$$$9^{{sin}^{2}x}=3^{\frac{3}{2}}$$$$3^{2{sin}^{2}x}=3^{\frac{3}{2}}$$$${sin}^{2}x=\frac{3}{4}$$$$sinx=\frac{\sqrt{3}}{2}\to x=\frac{\pi }{3}and(\pi -\frac{\pi }{3})i,e\frac{2\pi }{3}$$$$hencex=\frac{\pi }{6},\frac{\pi }{3},\frac{2\pi }{3},\frac{5\pi }{6}$$$$since\pi ⩾x⩾0sosinx\ne negetive$$$$sosinx\ne \frac{-\sqrt{3}}{2}$$$$$$

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