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 = (sin x)/(cos x) = (sin x)/(√(1-sin^2 x))`
where
`sin x = sin(sin^(-1) ((2)/(5))) = (2)/(5)`
Substituting back in gives:

`tan x = \frac{(2)/(5)}{\sqrt{1-((2)/(5))^2}} = \frac{(2)/(5)}{\sqrt{(21)/(25)}} = (2)/(5)*(√(25))/(√(21)) = (2)/(√(21))`


2) Use a right triangle.

`\theta = sin^(-1) ((2)/(5)) \\ \\ sin \theta = (2)/(5)`
sin = opp/hyp --> opp = 2, hyp = 5
Use Pythagorean theorem to solve for adjacent side.

`adj = √(5^2 - 2^2) = √(21)`
tan = opp/adj

`tan \theta = (2)/(√(21))`