Question 1205835: Solve the following equations for all x, unless the domain is restricted.
cos(1/2x) = 1/2
A. 2𝛑/3, + 2𝛑n
B. 2𝛑/3 + 2𝛑n, 7𝛑/6 + 2𝛑n
C. 7𝛑/6
D. no solution
E. 3𝛑/4 + 2𝛑n
F. 𝛑/2
Found 2 solutions by Theo, ikleyn:
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! 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.
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.
Answer by ikleyn(52919)  (Show Source):
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