Expert: Paul Klarreich - 9/14/2007

Question
I am not sure that you would be the right person to ask but I was just
wondering if you knew how I would simplify a problem such as:

tan(sin^-1(x))

I just started calc and I am confused on how to simplify it and I wasn't sure
who to ask because my teacher is away for the week. thanks if you can help
and thanks for your time even if you can't.

~Carrie

Answer
Questioner:   Cathrine
Category:  Calculus
Private:  No
 
Subject:  calc problem
Question:  I am not sure that you would be the right person to ask but I was just
wondering if you knew how I would simplify a problem such as:

tan(sin^-1(x))

I just started calc and I am confused on how to simplify it and I wasn't sure
who to ask because my teacher is away for the week. thanks if you can help
and thanks for your time even if you can't.

~Carrie
--------------------------
Hi, Carrie,

Here is how you do all of these examples:

An arcsin (arccos, arctan, etc) function represents an angle, so give that angle a name. (theta is an obvious choice)

[I write arcsin because (1) it doesn't make you think of a negative exponent and (2) it can be typed without using the shift key.]

So write @ = arcsin(x)   [Yes, @ is used to represent 'theta'.]

Then  x = sin @.

Now, remembering that  sine = opposite leg/hypotenuse, write:

sin @ = x/1,  and draw this right triangle:

     /|B
    / |
  1/  | x
  /   |
 /    |
A/@____+C

You see that since  sin @ = x/1, I have marked  BC = x, AB = 1.  Now it is easy to compute:

AC = sqrt(1 - x^2)

Finally,  tan @ = opp leg/adj leg
       x
= -------------
 sqrt(1 - x^2)