SOLUTION: if {{{cos(x) = -5/13}}} and {{{pi < x < 3pi/2}}}, find {{{cos(x/2)}}} using half angle identities

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Question 1202005: if cos%28x%29+=+-5%2F13 and pi+%3C+x+%3C+3pi%2F2, find cos%28x%2F2%29 using half angle identities
Answer by Edwin McCravy(20064) About Me  (Show Source):
You can put this solution on YOUR website!
The identity to use is cos%28x%2F2%29%22%22=%22%22%22%22+%2B-+sqrt%28%281%2Bcos%28x%29%29%2F2%29

Since we are finding cos%28x%2F2%29 we divide all three sides of the domain for
x by 2, so we'll have the domain for x%2F2.
 
pi+%3C+x+%3C+3pi%2F2, divide through by 2: 

pi%2F2+%3C+x%2F2+%3C+3pi%2F4, which in degrees is between 90o and 135o.

which means x/2 is in QII.  The cosine is negative in QII, so we use the
negative sign

cos%28x%2F2%29%22%22=%22%22-sqrt%28%281%2Bcos%28x%29%29%2F2%29%29

cos%28x%2F2%29%22%22=%22%22-sqrt%28%28%281%2B%28%22-5%2F13%22%29%29%2F2%29%29

cos%28x%2F2%29%22%22=%22%22-sqrt%28%281-%225%2F13%22%29%2F2%29

Multiply inside the square root radical by 13/13

cos%28x%2F2%29%22%22=%22%22-sqrt%28expr%28%28%281-%225%2F13%22%29%2F2%29%29%2813%2F13%29%29

cos%28x%2F2%29%22%22=%22%22-sqrt%28%2813-5%29%2F26%29

cos%28x%2F2%29%22%22=%22%22-sqrt%288%2F26%29

cos%28x%2F2%29%22%22=%22%22-sqrt%284%2F13%29

cos%28x%2F2%29%22%22=%22%22-2%2Fsqrt%2813%29

You can rationalize the denominator of that if you like.

Edwin