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/
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