For each angle in radians below determine the quadrant in wh

For each angle (in radians) below, determine the quadrant in which the terminal side of the angle is found. (a) theta = 10 pi/3 is found in quadrant (b) theta =-7 pi/4 is found in quadrant (c) theta = 1 pi/6 is found in quadrant (d) theta = 6 is found in quadrant

Solution

The formula for converting rads to degs is Degrees = Radians * 180/.

First, it might help to look at the Cartesian plane with the quadrants and degrees labelled:
........................90
........................|
.......II (-x,y).......|........I (x,y)
........................|
........................|
180----------------------------------...
........................|
........................|
.......III (-x,-y).....|........IV (x,-y)
........................|
........................270

Looking at the above picture, we can see that an angle whose angle is, say, 200 would have a terminal side lying in the third quadrant.

Now let\'s convert the angle in problem a) into degrees. a) = 10/3. Multiply by 180/:

10......180.......1800
____ * _____ = ______
...3....................3

In the above answer, the \'s cancel out (/ = 1) and 1800/3 = 600. We know that a circle is 360, so any number of degrees over this is just going around the circle more than once. If we subtract 360 from our answer (as many times as we need to, to get the answer less than 360). 600 - 360 = 240 so we can look at our picture of the coordinate plane above and see that 240 is between 180 and 270 so that terminal side lies in the III quadrant.

b)

-7......180.......-1260
____ * _____ = ______ = -315
...4....................4

Angle = -315 + 360 = 45 ======> 1st quadrant

c)

......180.......180
____ * _____ = ______ = 30
...6....................6

Angle = 30 ======> 1st quadrant

d)

6......180.......1080
____ * _____ = ______ = 343.8
...1...................

Angle = 343.8 is between 270 and 360 so that terminal side lies in the IV quadrant.