Expert: Paul Klarreich - 12/16/2010

Question
Use half-angle formula formula to find exact value of cos(x/2)when cosx= -(12/13), π<x<3π/2

I have no idea what to do and the teacher told me to figure it out using examples in the book which does not make any sense.

Answer
Questioner: Rohith
Country: United States
Category: Advanced Math
Private: No
Subject: Pre-Calculus
Question: Use half-angle formula formula to find exact value of cos(x/2)when cosx= -(12/13), π<x<3π/2

I have no idea what to do and the teacher told me to figure it out using examples in the book which does not make any sense.
..........................................
Start by WRITING a half-angle formula:

                   1 + cos x
cos(x/2) = +- sqrt(-----------)
                       2

Now analyze the angle, so you can decide on the +- thing; you do not get two answers, you get just one and you must select + or -.

Now if  π < x < 3π/2, then (divide by 2)

π/2 < x/2 < 3π/4

So x/2 is between 90 and 135 degrees; it is in the second quadrant. Since cosine is negative in the second quadrant, you must come out with a negative answer.

Let's get it:

                  1 - 12/13
cos(x/2) = - sqrt(-----------)
                      2

                  13 - 12
cos(x/2) = - sqrt(---------)
                    26

                  1
cos(x/2) = - sqrt(--)
                 26

That is it. (yes, that is an exact answer)