SOLUTION: Find all real numbers in the interval[0,2π) that satisfy the equation √2cos(x/2)-1=0.

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Question 760334: Find all real numbers in the interval[0,2π) that satisfy the equation √2cos(x/2)-1=0.
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
sqrt%282%29cos%28x%2F2%29-1=0 --> cos%28x%2F2%29=1%2Fsqrt%282%29=sqrt%282%29%2F2
The angles that have a cosine value of 1%2Fsqrt%282%29=sqrt%282%29%2F2 are in the first and fourth quadrant.

x%2F2=pi%2F4 is a solution in the first quadrant
That gives us:
x%2F2=pi%2F4 --> highlight%28x=pi%2F2%29
The next fist quadrant solution to the equation is
x%2F2=pi%2F4%2B2pi=9pi%2F4 --> x=9pi%2F2%3E2pi which is not in the interval [0,2pi)

x%2F2=-pi%2F4 is a fourth quadrant solution to sqrt%282%29cos%28x%2F2%29-1=0
and so are all angles differing by a multiple of 2pi
That makes x%2F2=-pi%2F4%2B2pi=7pi%2F4 a solution to sqrt%282%29cos%28x%2F2%29-1=0
However, x%2F2=7pi%2F4 --> x=7pi%2F2%3E2pi which is not in the interval [0,2pi).

SO the only solution in the interval [0,2pi) is highlight%28x=pi%2F2%29.