SOLUTION: Express tan(arcsin(x)) without trig functions.

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Question 612706: Express tan(arcsin(x)) without trig functions.
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
arcsin(x) represents the angle whose sin is x. So tan(arcsin(x)) represents the tan ratio for that angle.

It will be easier to understand if you...
  1. Draw a right triangle.
  2. Pick one of the acute angles (i.e not the right angle) to be the arcsin(x) angle. Let's call this angle A.
  3. Since the sin ratio is opposite/hypotenuse and since we want the sin ration to be x, label the side opposite to A as "x" and the hypotenuse as "1". It should be clear that the sin ration for A is x/1 = x.
    Since we want the tan ratio for angle A and since the tan ratio is opposite/adjacent, we need the adjacent side to angle A. Let's label the adjacent side "y".
  4. Use the Pythagorean theorem and solve for y:
    x%5E2+%2B+y%5E2+=+1%5E2
    x%5E2+%2B+y%5E2+=+1
    y%5E2+=+1-x%5E2
    y+=+sqrt%281-x%5E2%29
  5. Write the tan ratio for A, using the square root expression for the adjacent side:
    tan(arccos(x)) = x%2Fsqrt%281-x%5E2%29
    This may be an acceptable solution to your problem. But...
  6. Our expression for tan(arccos(x)) has a square root in its denominator. Usually these denominators get rationalized:
    tan(arccos(x)) =