What the coordinate of the terminal point of 19 pi

1
by angelajm

2015-02-17T09:49:36+08:00
The terminal point is where you land on a unit circle if you travel around (counterclockwise, starting at (1,0)) for t radians.

t = 19pi/6 = (12pi + 7pi)/6 = 2pi + 7pi/6

The distance around a circle is only 2pi, so we can throw out one whole rotation and call it 7pi/6.

For pi/6, the terminal point is (sqrt3/2, 1/2). It's one of the standard identities to learn to either derive or memorize whenever it comes up.

For 7pi/6, it's pi/6 then a full half rotation (pi), which will swing it down from +x,+y to -x,-y but not change anything else, so...

P(-sqrt3/2, -1/2)

The reference number is what you get when you shave off all the extra 2pi and pi and pi/2, so all you have left is the remainder in the first quadrant (+x,+y). You use the extra round values of pi to move around the circle and see which quadrant the answer will land in. Then you flip around + and - as necessary to jump from the reference number to the correct quadrant.

In this case, the reference number is pi/6.
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