Expert: Sherry Wallin - 11/21/2010

Question
Explain the reason,why -5pi/9 and 5pi/9 are not coterminal angles.
Dear Ms.Sherry,
So, my textbook says that two angles are coterminal if they differ in measure by a multiple of 2pi or 360. So, according to that I am supposed to state the reason why the above two angles are not coterminal. So,I gave it as follows: 5pi/9 - (-5pi/9) = 10pi/9.
But Isn't 10pi/9 a multiple of 2pi.Because,see 2pi(5/9).So it does have 2pi common.Can you please tell me whether I am explaining correctly or not,because I dont think so i am illustrating it properly.Can you please explain where I am making mistake.
                               Thanks a lot,
                                Bulbul.

Answer
Hi Bulbul~

If they differ means you add/subtract a multiple of 2pi to your angle. In your case this is: -5pi/9 + 18pi/9(this is 2pi) = 13pi/9 or
-5pi/9 - 2pi = -5pi/9 - 18pi/9 = -23pi/9 and the other angle is:
5pi/9 + 18pi/9 = 23pi/9 or 5pi/9 - 18pi/9 = -13pi/9

The other way to see this is that 5pi/9 is in the 2nd quadrant, i.e., it is between pi/2 and pi. pi/2 = 4.5pi/9 or 9pi/18 ans 5pi/9 is 10pi/9 thus pi/2 < 5pi/9 < pi whereas -5pi/9 is in the 3rd quadrant, ie., it is between -pi/2 and -pi since -pi/2 = -9pi/18 and
-pi = -18pi/-18 and -5pi/9 = -10pi/18 thus -18pi/18 <-10pi/18< -9pi/18

Math Prof

What you did was take the difference of the two given angles and tried to show that they had a multiple of 2 pi in them but what you want to show is that -5pi/9 = +-2pi*n*5pi/9