help me write an algebraic expression for tan(arcsin 2x/3)?

Answer 1

tan (arcsin(2x/3))

To solve this:

Let t = arcsin(2x/3)
{Side note: In the end, you'll want to solve for tan(t), because
tan(t) = tan(arcsin(2x/3)}.

Then, if we take the sine of both sides, the sine and the arcsine will cancel each other out, leaving just 2x/3. So we have

sin(t) = 2x/3

At this point, we note that, for a right angle triangle,
sin(t) = opp/hyp {by the SOHCAHTOA trig rules}. That means

sin(t) = opp/hyp = 2x/3, so

opp = 2x, hyp = 3.
adj can be calculated using the Pythagorean theorem, and it would be sqrt(hyp^2 - opp^2), or
sqrt ( 3^2 - [2x]^2) = sqrt(9 - 4x^2).

opp = 2x
hyp = 3
adj = sqrt(9 - 4x^2).

Therefore, given that sin(t) = 2x/3, then
tan(t) = opp/adj = 2x / sqrt(9 - 4x^2)