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### 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

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
```
Associated Topics:
High School Trigonometry

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