Trig Inverses: sin(arctan x)

Date: 11/04/2002 at 15:06:32
From: Erin
Subject: Trig Inverses

We got a question in my trig class today, and I don't know where to 
start. We discussed the inverses of trig functions, but when the 
teacher threw an x in there, I didn't understand. Here is the problem 
she gave us.

sin(arctan x)

Thanks for all your help.

Date: 11/05/2002 at 14:05:41
From: Doctor Ian
Subject: Re: Trig Inverses

Hi Erin,

When you have an expression like 

  sin(x) 

it means that if you supply the value of an angle, the function will
return a corresponding value:

  sin(45 degrees) = sin(pi/4 radians) = sqrt(2)/2

So if you have

  sin( f(x) )

it implifes that f(x) is yet another kind of function, that takes some
value of x and returns an angle, which will then be the argument to
the sine function. 

Does that make sense so far? 

Well, arctan is just such a function. It takes a value, such as 1, and 
returns the angle that _would_ give you that value, if used as an 
input to the tangent function.  So

  1)  tan(45 deg) = 1

  2)  arctan(1) = 45 deg
 
  3)  tan( arctan(1) ) = tan( 45 deg ) = 1

  4)  arctan( tan( 45 deg) ) = arctan(1) = 45 deg

The other common name for 'arctan' is 'inverse tangent', and these
examples show why: the tan() and arctan() functions cancel each other
out (although you have to be careful to keep track of what quadrant
you're talking about).  

The expression 

  sin(arctan x)

means

  give me some value; arctan will tell you for which angle
  tan(x) = that value; and sin(...) will tell you the sine
  of the angle. 

Let's work through it for the following triangle,

    |\
  a | \ c 
    |__\
     b  A

using x = (a/b) for our argument:

    sin( arctan (a/b) )

  = sin( A )                 because tan(A) = a/b,
                             so arctan(a/b) = A

  = a/c                      because sin(A) = a/c
 
Does this help? 

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/