Rearranging x=tan()sin()

Date: 05/01/2002 at 21:46:12
From: Carl Segor
Subject: Rearranging x=tan()sin()

How would I approach rearranging:

   x = tan(y)*sin(y)

into

   y = some function of x

Date: 05/01/2002 at 23:20:01
From: Doctor Jubal
Subject: Re: Rearranging x=tan()sin()

Hi Carl,

Thanks for writing to Dr. Math.

An intermediate goal here is going to be to be able to write 
something of the sort

  trig(y) = f(x)

and then taking the inverse trig function of both sides will give us 
an answer. So we need to get everything in terms of the same trig 
function.

The tangent is defined in terms of sines and cosines, so we could 
begin by writing

  x = [sin(y) / cos(y)] * sin(y)

  x = sin^2(y) / cos(y)

Now at this point, we need to get everything in terms of all sines or 
all cosines. I think the easiest way to proceed is to use the 
Pythagorean identity:

  sin^2(y) + cos^2(y) = 1

to get everything in terms of cosines.  Since sin^2(y) = 1 - cos^2(y), 
we can now write

  x = [1 - cos^2(y)] / cos(y)

This gets us to our goal of having everything in terms of one trig 
function, but not to the goal of having something like trig(y) = f(x).  
But notice we can rearrange this to

  cos(y) * x = 1 - cos^2(y)

  cos^2(y) + x cos(y) - 1 = 0

Which is a quadratic function of cos(y). So now, you could use the 
quadratic formula to solve for cos(y), and then taking the arccos of 
both sides will give you your answer.

Does this help?  Write back if you'd like to talk about this some
more, or if you have any other questions.

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

Date: 05/02/2002 at 11:43:08
From: Carl Segor
Subject: Rearranging x=tan()sin()

Dr. Jubal,

Thanks for the insight. The conclusion is near!

Regards,
Carl