How do I find the formula for sin(tan^-1(x/sqrt(25-x^2)))

Answer 1

It is x/5.

I am using capitals for the tan and sin definition so that we don't get confused with the multiple x's floating around.

tan = Y/X, and sin = Y/R

Then, if tan = x/sqrt(25-x^2), you can see that Y = x, and X = sqrt(25-x^2), so R = sqrt(25) = 5

Then, sin = Y/R = x/5

I would have preferred in the problem if they had used u or a instead of x. To see that the result makes sense, let x = 3, and you will get the well-known 3-4-5 right triangle.