Question 1181850
i used x instead of theta.
equation is:
4 * cos(2x) + 1 = 2
subtract 1 from both sides to get:
4 * cos(2x) = 1
divide both sides by 4 to get:
cos(2x) = 1/4
solve for 2x to get:
2x = arccos(1/4) = 75.52248781
solve for x to get:
x = 37.76124391 degrees.


since the frequency is 2, the period is 360/2 = 180.
that means two full cycles of the cosine function in 360 degrees.
when the period is 180 degrees, the 4 quadrants are:
0 to 45, 45 to 90, 90 to 135, 135 to 180.
the cosine function is positive in the first and fourth quadrant.
the angle is 37.76124391 in the first quadrant and 180 minus 37.76124391 in the fourth quadrant = 142.2387561.
in the second 180 degree period, the angle in the first quadrant of that = 37.76124391 + 180 = 217.7612439 degrees and the angle in the fourth quadrant of that = 142.2387561 + 180 = 322.2387561.


the graph of the two equations of y = 4 * cos(2x) + 1 and y = 2 is shown below.
since 4 * cos(2x) + 1 = 2, the intersection of thoose two equations is the solution.
the graph repeats endlessly in either direction, but the focus is on the interval between 0 degrees and 360 degrees.


<img src = "http://theo.x10hosting.com/2021/061301.jpg" >


the general form of the cosine equation is:


y = a * cos(b * (x-c)) + d
a is the amplitude
b is the frequency
c is the horizontal shift
d is the vertical shift


in the equation of y = 4 * cos(2x) + 1, .....
a = 4
b = 2
c = 0
d = 1


to graph the equation, i basically broke up the one equation of 4 * cos(2x) + 1 = 2 into 2 equations.
they were y = 4 * cos(2x) + 1 and y = 2.
the intersection of those 2 equations is where the solution was.