SOLUTION: How do I develop the identity for {{{sin(expr(1/2)a)}}} by using the identity for {{{cos (2a)}}}?

Algebra ->  Trigonometry-basics -> SOLUTION: How do I develop the identity for {{{sin(expr(1/2)a)}}} by using the identity for {{{cos (2a)}}}?      Log On


   

Question 451184: How do I develop the identity for sin%28expr%281%2F2%29a%29 by using the identity for cos+%282a%29?
Answer by Edwin McCravy(20081) About Me  (Show Source):
You can put this solution on YOUR website!

sin%28expr%281%2F2%29a%29

Start with the identity:

Cos%282alpha%29+=+Cos%5E2alpha+-+Sin%5E2alpha

Cos%282alpha%29+=+%281-Sin%5E2alpha%29+-+Sin%5E2alpha

Cos%282alpha%29+=+1-Sin%5E2alpha+-+Sin%5E2alpha

Cos%282alpha%29+=+1-2Sin%5E2alpha <-- sometimes this is given. If so you can start here

2Sin%5E2alpha+=+1+-+Cos%282alpha%29

Sin%5E2alpha+=+%281+-+Cos%282alpha%29%29%2F2

Sin%28alpha%29+=+%22%22+%2B-+sqrt%28%281+-+Cos%282alpha%29%29%2F2%29

Now substitute alpha=expr%281%2F2%29a

Sin%28expr%281%2F2%29a%29+=+%22%22+%2B-+sqrt%28%281+-+Cos%282expr%281%2F2%29a%29%29%2F2%29

Do one cancellation on the right:



Sin%28expr%281%2F2%29a%29+=+%22%22+%2B-+sqrt%28%281+-+Cos%28a%29%29%2F2%29

The sign to use depends on the quadrant
in which the terminal side of expr%281%2F2%29a lies. 

If the terminal side of expr%281%2F2%29a lies in the 
first or second quadrants, the identity uses the 
positive sign.

If the terminal side of expr%281%2F2%29a lies in the 
third or fourth quadrants, the identity uses the 
negative sign.

Edwin