Why do we have to add 180 degree when getting ARCTAN results from calculator?

Question

In some Physics questions, whenever we get results from the calculator denoting the arctan, we often have to either subtract or add 180 or 90 degrees from the result.

HOW THEN, CAN I KNOW WHEN TO ADD OR SUBTRACT 180 or 90 DEGREES FROM THE CALCULATOR's RESULT?

Please answer this question. THere are many of us depending on this question. Thanks a lot!

Ten points goes to the best answerer.

Answer 1

You need the arctan function arctan(a) when you have a ratio a = y / x and you want to know what angle this represents.

But the calculator has a problem: each ratio can represent two different angles.

Suppose a = 1. If both x and y are positive, this represents an angle of 45 deg. Fine, but what if they are both negative? You still get a = 1, but now the angle is 225 deg. How is the poor calculator to know? You haven't told it what x and y are, only what their ratio is.

So what the calculator does is give the answer you would get if x were positive. If x is negative, it's up to you to add 180 degrees to the result.

So the rule is: look at the actual x and y that give you the ratio you are feeding to the calculator. If x is positive, just use the result you get. If x is negative, add 180 degrees.

Hope this helps.

Answer 2

Arctan In Degrees

Answer 3

Your calculator assumes TAN(theta) = abs(y/x) so ATAN(abs(y/x)) = theta. In other words -y/x = -y/-x = y/-x = y/x as far as the calculator is concerned. And, of course, we know that isn't true.

y/x: puts the angle in the first quadrant (0 to 90 deg)
y/-x: puts the angle in the second (90 to 180)
-y/-x: puts it in the third (180 to 270)
-y/x: puts it in the fourth (270 to 360)

By convention, all the degrees (quadrants) rotate in the CCW (counterclockwise) direction. So you need to take the signs of y and x into account when reading off your answers in degrees.

For example, suppose you enter y = 4 and x = -4; so your calculator comes up with 45 deg. or pi/4 if you have it set to radians.

But 45 deg is in the first quadrant and the y = 4 and x = -4 clearly puts it in the second quadrant. The second quadrant is 90 deg further away than the first quadrant; so we need to rotate the angle the calculator gave us by 90 degrees CCW. That means we have to add 90 deg to get theta = theta given by calculator + 90 deg rotation = 45 + 90 = 135 deg measured CCW from the plus x axis.

Similarly, if your input were -y/-x, you'd need to rotate your calculator answer by 180 degrees CCW because -y/-x is in the third quadrant. And -y/x would require 270 deg rotation CCW.

Answer 4

ATAN is a multvalued function, but a calculator only gives you the principle value. One usually takes the ATAN of a ratio of two numbers Y/X to determine a phase angle. You need to consider the sign of Y and X separately, however, to determine which quandrant of the phase diagram you are in.

For some applications, you have to track the phase over several orders, so you also have to keep track of the *order* of the angle. The tangent of an angle is the same if you add N times 360 degrees to it, where N is an integer. So, 360 + ATAN is the angle's ATAN too!

Answer 5

The answer to your question was explained very well on math.stackexchange.com by a user named "Carl Mummert". I've copied & pasted his response:

"The general issue is that the tangent function has a period of 180 degrees, but there are 360 degrees in a circle. Therefore the arctangent function cannot, in general, tell the correct angle when you convert from cartesian coordinates to polar coordinates, because for any point (x,y) the point (−x,−y) will lead to the same value of arctangent when you use the formula θ=arctan(y/x).

So, when converting from cartesian to polar, you have to use additional knowledge of which quadrant the point is in, which allows you to "fix" the value of arctangent to give the correct angle. You can do this by adding 180 degrees (π radians) to the angle from the formula if it is in the opposite quadrant from the one you want."

Answer 6

The main values are between -pi/2 and pi/2...the entry being a TAN of an angle...anyway it is about "some" or "few" of you, not "we"...