SOLUTION: please Help! Rewrite the expression as an algebraic expression in x. sin(tan^{-1}(13x)

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Question 424858: please Help!
Rewrite the expression as an algebraic expression in x.
sin(tan^{-1}(13x)
Answer by jsmallt9(3758)   (Show Source): You can put this solution on YOUR website!
sin(tan-1(13x))
This expression says "the sine of the angle whose tangent is 13x. To understand how we get an answer to your problem it will help greatly if we have a diagram to look at. So
  1. Draw a right triangle. (It doesn't matter what it looks like. Any right triangle will do.)
  2. Pick one of the acute angles and label it A. (It doesn't matter which angle or how big or small the angle looks.)
  3. Label the side opposite angle A as 13x.
  4. Label the side adjacent to angle A as 1.
  5. Label the hypotenuse as h.
Knowing that tan is we should be able to see that tan(A) = or just 13x. So A is tan-1(13x). Since sin is , sin(A) = . Now we just need to find the hypotenuse. We can find the hypotenuse, in terms of x, using the Pythagorean Theorem:

Solving...


So now
sin(tan-1(13x)) = sin(A) =
This may be an acceptable answer. But often answers with square roots in the denominator are not acceptable. So we will go ahead and rationalize the denominator:
sin(tan-1(13x)) = sin(A) =
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