Question

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

Answer 1

There are 2 ways to do this:

1) Transform tan into terms of sin.
$tan x = \frac{sin x}{cos x} = \frac{sin x}{\sqrt{1-sin^2 x}}$
where $sin x = sin(sin^{-1} (\frac{2}{5})) = \frac{2}{5}$
Substituting back in gives:
$tan x = \frac{\frac{2}{5}}{\sqrt{1-(\frac{2}{5})^2}} = \frac{\frac{2}{5}}{\sqrt{\frac{21}{25}}} = \frac{2}{5}*\frac{\sqrt{25}}{\sqrt{21}} = \frac{2}{\sqrt{21}}$


2) Use a right triangle.
$\theta = sin^{-1} (\frac{2}{5}) \\ \\ sin \theta = \frac{2}{5}$
sin = opp/hyp --> opp = 2, hyp = 5
Use Pythagorean theorem to solve for adjacent side.
$adj = \sqrt{5^2 - 2^2} = \sqrt{21}$
tan = opp/adj
$tan \theta = \frac{2}{\sqrt{21}}$