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### Arctan and Polar Coordinates

```
Date: 03/09/99 at 14:22:44
From: Jim
Subject: What is an "arctan"?

As the last step in solving an example, we change 7.36R + 26.31U to
polar form.

H = sqrt[(97.36)^2 + (26.31)^2] = 27.32
tan (th) = 26.31/7.36 = 3.575
th = arctan 3.575 = 74.37 deg

answer:  27.32 /_74.37 deg

What is an arctan and how did it do that?
```

```
Date: 03/09/99 at 17:22:02
From: Doctor Rick
Subject: Re: What is an "arctan"?

Let me guess what the problem means: 7.36 units to the right and 26.31
units up. You could draw this on a graph as (x, y) = (7.36, 26.31),
that is, x = 7.36 and y = 26.31. Then you want to find the polar
coordinates of this point: the distance of the point from the origin
(0, 0), and the angle that the line from (0, 0) to (x, y) makes
with the positive x axis.

What you call H is the distance from the origin, or radius, of the
point. You did not ask about it, so I guess you realize that this
equation is the Pythagorean Theorem:

r = sqrt(x^2 + y^2)

Now for the arctan. Do you remember what a tangent is? Draw a line from
the point (x, y) perpendicular to the axis. This line, the x axis, and
the line from (0, 0) to (x, y) form a right triangle:

(x, y)
/|
/ |
/  |
/   |
/    | y
/     |
/      |
/ th    |
/________|...........> (x axis)
(0, 0)   x    (x, 0)

The tangent of angle theta is the ratio of the opposite side of the
triangle to the adjacent side, or y/x. The arctangent, also called the
inverse tangent, of y/x is the angle theta that has this tangent. That
is what is going on in your example.

theta = arctan(y/x)

This works in quadrant I (right and up) where the angle is positive,
and in quadrant IV (right and down) where the angle is negative,
because the arctan function returns a value between -pi/2 and pi/2
radians, or between -90 and 90 degrees. But if both x and y are
negative (left and down), y/x is the same as if they were both
positive. Also, we must be careful about what happens if x = 0, since
we cannot divide by 0. So the rule for converting to polar coordinates
has to be more complex:

if x > 0 then
theta = arctan(y/x)
if x < 0 then
theta = arctan(y/x) - 180 degrees
if x = 0 then
if y > 0 then
theta = 90 degrees
if y < 0 then
theta = -90 degrees
if y = 0 then
theta is indeterminate

Many computer languages have a function called atan2(y, x) that does
all this automatically. Many scientific calculators have a R -> P key
(rectangular to polar) that does the whole thing automatically, finding
the radius as well as the angle.

I hope this helps you understand the method you are learning. Keep
asking good questions like this!

- Doctor Rick, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Trigonometry

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