Proving Identity for Sum of CosinesDate: 10/17/2003 at 22:01:36 From: Amanda Subject: The Sum of Cosines How do you prove the identity for the sum of cosines? What are some good problems to test understanding of this subject? Date: 10/18/2003 at 07:13:29 From: Doctor Luis Subject: Re: The Sum of Cosines Hi Amanda, Proving the identity for sum of cosines is somewhat tricky. Here's how you do it. First, you look at the sum of two cosines cos(A) + cos(B) and next, you introduce new variables x and y, defined in terms of A and B, cos(A) + cos(B) = cos(x + y) + cos(x - y) Now, you apply the cosine addition formula to the right hand side of that last equation cos(x + y) + cos(x - y) = (cos(x)cos(y)-sin(x)sin(y)) + (cos(x)cos(y)+sin(x)sin(y)) = 2cos(x)cos(y) We're almost done. We just have to express x and y in terms of A and B. A = x + y B = x - y You can see that A + B = 2x, and A - B = 2y. This means x = (A + B)/2 y = (A - B)/2 And that proves the sum of cosines formula cos(A) + cos(B) = 2 cos((A+B)/2) cos((A-B)/2) To see if you understand this process, see if you can come up with identities for sin(A) + cos(B) sin(A) + sin(B) sin(A) - sin(B) cos(A) - cos(B) sin(A) - cos(B) Those would be good problems to do. I hope this helped! Let us know if you have any more questions. - Doctor Luis, The Math Forum http://mathforum.org/dr.math/ |
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