Question 1205835
i believe tis can be solved as follows:


start with cos(1/2 * x) = 1/2


i'll work in degrees because it's easier.
you can then convert to radians by using the formula degrees = radians * 180 / pi.


solve for the angle by taking arcos(1/2) = 60 degrees.
that's in the first quadrant.
cosine is positive in the first and the fourth quadrant.


in the fourth quadrant, the equivalent angle is 360 - 60 = 300 degrees.


you get 1/2 * x = 60 degrees and 300 degrees.
using your calculator, you find that:
cos(60) = 1/2
cos(300) = 1/2


solving for x, you get x = 120 degrees and 600 degrees.


converting from degrees to radians, you get x = 120 * pi / 180 = 2/3 * pi and you get x = 600 * pi / 180 = 10/3 * pi.


x = 2/3 * pi and 10/3 * pi.


since the frequency is 1/2, then the period = 2pi / (1/2) = 4pi.


the cosine function will repeat every 4pi units forever going forward and backward.


consequently, your solution is x = 2/3 * pi plus or minus 4pi * n and 10/3 * pi plus or minus 4pi * n.


on a graph, it would look like this.


<img src = "http://theo.x10hosting.com/2024/012641.jpg">


the equation for the graph is y = cos(1/2 * x)


i only showed 3 periods:
the period before
the base period
the period after


the solution is x = 2/3 * pi plus or minus 4pi * n and 10/3 * pi plus or minus 4pi * n.


in the base period you get x = 2pi/3 and 10pi/3


in the period before you get x = 2pi/3 - 12pi/3 = -10pi/3 and you get 10pi/3 - 12pi/3 = -2pi/3.


in the period after you get 2pi/3 + 12pi/3 = 14pi/3 and you get 10pi/3 + 12pi/3 = 22pi/3.


note that 12pi/3 is equal to 4pi so that plus or minus 4pi is the same as plus or minus 12pi/3.


unfortunately 2pi/3 and 10pi/3 are not in the solution set shown.


if the solution has to be one of the ones shown, then the solution would have to be selection D which should be re-labeled as solution not shown, because there is a solution but it's not shown.