Mathematics
DinahParmar116
23

Find the exact value of tan (arcsin (two fifths))

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(1) Answers
brookie

There are 2 ways to do this: 1) Transform tan into terms of sin. [latex]tan x = \frac{sin x}{cos x} = \frac{sin x}{\sqrt{1-sin^2 x}}[/latex] where [latex]sin x = sin(sin^{-1} (\frac{2}{5})) = \frac{2}{5}[/latex] Substituting back in gives: [latex]tan x = \frac{\frac{2}{5}}{\sqrt{1-(\frac{2}{5})^2}} = \frac{\frac{2}{5}}{\sqrt{\frac{21}{25}}} = \frac{2}{5}*\frac{\sqrt{25}}{\sqrt{21}} = \frac{2}{\sqrt{21}}[/latex] 2) Use a right triangle. [latex]\theta = sin^{-1} (\frac{2}{5}) \\ \\ sin \theta = \frac{2}{5}[/latex] sin = opp/hyp --> opp = 2, hyp = 5 Use Pythagorean theorem to solve for adjacent side. [latex]adj = \sqrt{5^2 - 2^2} = \sqrt{21}[/latex] tan = opp/adj [latex]tan \theta = \frac{2}{\sqrt{21}}[/latex]

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