Question 909017
there are 4 quadrants in the interval from 0 to 2pi.
first quadrant is 0 to pi/2 (same as 0 to 90 degrees)
second quadrant is pi/2 to pi (same as 90 degrees to 180 degrees)
third quadrant is pi to 3pi/2 (same as 180 to 270 degrees)
fourth quadrant is 2pi(same as 270 to 360 degrees)


once you get past 2pi, the cycle repeats itself.


your solutions will be in the interval of 0 to 2pi and then they will repeat themselves.


let's do you problem and you'll see what i mean.


if you use your calculator to find the angle whose sine is -.2, the calculator will give you an angle of -.2013579208 radians.


that angle is in the fourth quadrant.


that doesn't really help you too much.


it's better to find the angle in the first quadrant.


you do that by making the sine positive since all trig functions are positive in the first quadrant.


when you solve for sinx = .2, you get an angle of .2019579208 radians.


that becomes your reference angle.


you can use that angle to find all the other angles.


now you know the sine is negative because that what you were given.


the sine is negative in quadrants 3 and 4.


the rules for finding an angle in each quadrant are as follows:


your reference angle is in quadrant 1.


the rules for finding the angle in each quadrant are as follows:


the angle in quadrant 1 is the reference angle.
the angle in quadrant 2 is pi minus the reference angle.
the angle in quadrant 3 is pi plus the reference angle.
the angle in quadrant 4 is 2pi minus the reference angle.


you want the angle in quadrant 3 and quadrant 4.


your angle in quadrant 1 is .2019579208 radians which is your reference angle.


your angle in quadrant 3 will be pi + .2019579208 radians =  3.342950574 radians


your angle in quadrant 4 will be 2pi - .2019579208 = 6.081827386 radians


your angle will be multiples of plus or minus 2pi radians until you get past the interval you are interested in.


the interval you are interested in is -pi radians to 5pi radians.


-pi radians is equal to -3.142592654 radians.


5pi radians is equal to 15.70796327 radians.


your interval of interest is from -3.142592654 radians to 15.70796327 radians.


take your angle in quadrant 2 and keep adding 2*pi until you get past 15.70796327 radians


you will get:


3.342950574 + 1*2pi = 9.626135882 radians
3.342950574 + 2*2pi = 15.90932119 radians ***** > 15.70796327 radians


take your angle in quadrant 2 and keep subtracting 2*pi radians until you get past -3.142592654 radians.


3.342950574 - 1*2pi = -2.940234733 radians.
3.342950574 - 2*2pi = -9.22342004 radians ***** < -3.142592654 radians


take your angle in quadrant 4 and do the same.


6.081827386 + 1*2pi = 12.36501269 radians
6.081827386 + 2*2pi = 18.648198 radians ***** > 15.70796327 radians


6.081827386 - 1*2pi = -.2013579208 radians
6.081827386 - 2*2pi = -6.484543228 radians ***** < -3.142592654 radians


your solutions are:


-2.940234733 radians
-.2013579208 radians
3.342950574 radians
6.081827386 radians
9.626135882 radians
12.36501269 radians


this will be easier to see on a graph.
any difference in the numbers you see on the graph and the numbers shown above are due to rounding.


your graph will look like this:


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