Question

Write the expression sin(tan^-1)x) as an algebraic expression in x (without trig or inverse trig functions).

A. (sqrt (1+x^2))/ (x)
B. (1) / (sqrt (1+x^2))
C. (x) / (sqrt (1-x^2))
D. (x) / (sqrt (1+x^2))
E. (1) / (sqrt (1-x^2))

Answer 1

I'm liking answer D, although I wouldn't bet my life on it. See if you like my reasoning, and if not then by all means, show me where I lost it.

We are given sin[tan^-1(x)]. So the first thing I did was break it into two problems. We need to somehow write sin (theta), and theta = arctan x. If theta = arctan x, then tan (theta) = x. One of the shortcuts for remembering tan is opp/adj. If opp/adj is equal to x, then it is x/1. Drawing a right triangle from the origen of a circle and filling in the vertical leg as x, and the horizontal as 1, we can now use Pythags Theorem to fill in the hyp (hopefully arriving at sqrt(x^2 + 1). With all three lengths of our triangle now filled out, we can answer the next part of the question (regarding sin theta).

One of the shortcuts for remembering sin theta is opp/hyp. Pulling those values from our newly drawn triangle should give us x/sqrt(x^2 +1). Unless there is a flaw in my reasoning, I think D is your answer.

Answer 2

Here's @Ajk's reasoning with a picture: