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When it comes to..Coterminal Angles

We just learned Coterminal Angles today and i'm not sure how to make sure they are equal.
The question is 31pi divided by 18 and the other half of it is negative 67pi divided by 18.. I have to make sure they are equal..
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1 Answer

They aren't coterminal, but they would be is they were both the same sign. There are a number of ways to do this - you do have to remember that a circle is 2pi radians. You don't necessarily have to convert to degrees but sometimes it helps to visualize the situation.
2pi can be thought of as (36/18)pi, and you're given the angle (31/18)pi - well, your angle then has to be 31/36ths of a circle, or +310°, which can also be expressed as -50°.
(Another way to think of this is to divide 31pi by 18 to get 1.722pi, and then divide 1.722pi by 2pi to get the fraction of the circle (0.8611) and then multiply by 360° giving you 310° again)
Similarly, (-67/18)pi has to be -67/36ths of a circle, or -670°, which is best thought of as one complete turn "backwards" (-360°) plus another -310°, which can also be expressed as +50°
(The alternate way would be -67pi divided by 18 yielding -3.722pi, which is -1.8611 circles or -670° again)
So the first value is in the lower right quadrant, while the second is in the upper right quadrant - they're reflected in the x-axis. In general you remove multiples of 2pi from the values, and then compare them. In this case that converts the second angle to (-31/18)pi, the opposite of the first angle, and the only way two opposites can be coterminal is if they're 180 degrees, or +pi and -pi.
Coterminal angles for (31/18)pi would be (-5/18)pi, (-41/18)pi, or (67/18)pi
Coterminal angles for (-67/18)pi would be (5/18)pi, (41/18)pi, or (-31/18)pi
Hope this helps!