You can put this solution on YOUR website! i find it easier to work in degrees, so i usually convert everything to degrees and then convert back to radians at the end.
once you work with radians for a while and feel more comfortable with them, then the conversion is not necessary as much.
i still have to convert to degrees to make sure i'm doing it right.
the conversion formulas are:
degrees = radians * 180 / pi
radians = degrees * pi / 180
your period from 0 to 2pi radians is equivalent to from 0 to 360 degrees.
start with:
3 + cos(2x) = 7/2
subtract 3 from both sides of the equation to get:
cos(2x) = 1/2
solve for 2x to get 2x = 60 degrees.
cosine is positive in quadrants 1 and 4.
at this point in time, i find it easier to work with 2x rather than converting to x right away.
i solve for 2x and work with that until everything is set up properly and then convert to x.
quadrant 4 degrees are equal to 360 degrees minus quadrant 1 degrees.
that makes quadrant 4 degrees equal to 360 - 60 = 300 degrees.
you have 2 occurrences of 2x between 0 and 360 degrees.
they are:
2x = 60 degrees and 300 degrees.
now its time to convert to x.
x/2 = x which means that x is equal to 30 and 150.
here's the tricky part.
the period of 2x is equal to 2pi radians which is equal to 360 degrees.
this makes the period of x equal to 2pi / 2 which is equal to pi if you are working in radians.
this makes the period of x equal to 360 / 2 = 180 degrees if you are working in degrees.
the conversion formula is stated as follows.
period = 2pi / frequency if you are working in radians.
period = 360 / frequency if you are working in degrees.
since we are working in degrees, we have the period of x being equal to 360 / 2 = 180 degrees.
a period of 180 degrees means that 2 full periods have been completed in 360 degrees.
so x is equal to 30 degrees and 150 degrees in one full period.
between 0 and 360 degrees you have 2 periods.
30 and 150 are in the first period.
add 180 to each of them and you get:
210 and 330 are in the second period.
your angles between 0 and 360 are:
30, 150, 210, 330.
that is when the graph of y = 3 + cos(2x) intersects with the graph of y = 7/2
that's also when the graph of y = 3 + cos(2x) - 7/2 intersects with the graph of y = 0.
here's a graph of your equations.
you can graph both equation separately as shown in the top graph in the picture, or you can subtract one equation from the other as shown in the bottom graph in the picture.
you will get the same answer both times.
the difference is where the intersection occurs.
in the top graph it occurs at y = 7/2.
in the bottom graph it occurs at y = 0.
y = 0 is the same as the x-axis.
you can now convert back to radians if you desire.
30 degrees is equivalent to pi/6 radians.
300 degrees is equivalent to 5pi/6 radians.
210 degrees is equivalent to 7pi/6 radians.
330 degrees is equivalent to 11pi/6 radians.
