Lillie Pittman

2022-07-17

How do you find the exact value of $\mathrm{tan}\left({\mathrm{sin}}^{-1}\left(0.1\right)\right)$?

bardalhg

Expert

Let $\theta ={\mathrm{sin}}^{-1}\left(0.1\right)⇒\mathrm{sin}\left(\theta \right)=0.1=\frac{1}{10}=\frac{Opposite}{Hypotenuse}=10$
By Pythagorean Theorem,
$\left(Adjacent\right)=\sqrt{{10}^{2}-{1}^{2}}=\sqrt{99}=3\sqrt{11}$
Hence,
$\mathrm{tan}\left({\mathrm{sin}}^{-1}\left(0.1\right)\right)=\mathrm{tan}\theta =\frac{Opposite}{Adjacent}=\frac{1}{3\sqrt{11}}$

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