OpenStudy (anonymous):

write the sum as a product: sin5x+sin3x

3 years ago
OpenStudy (aum):

\( \sin(A+B) = \sin(A)\cos(B) + \cos(A)\sin(B) \\ \sin(A-B) = \sin(A)\cos(B) - \cos(A)\sin(B) \\ \text{Add both sides: } \\ \sin(A+B) + \sin(A-B) = 2\sin(A)\cos(B) \\ \) Set A + B = 5x and A - B = 3x. Solve for A and B and plug it into the equation on the last line above.

3 years ago
OpenStudy (aum):

A+B = 5x --- (1) A-B = 3x --- (2) add both sides: 2A = 8x A = 4x From (1): 4x + B = 5x B = x. sin5x+sin3x = sin(4x+x) + sin(4x-x) From the derivation earlier: sin(A+B) + sin(A-B) = 2sin(A)cos(B) Therefore, sin5x+sin3x = sin(4x+x) + sin(4x-x) = 2sin(4x)cos(x)

3 years ago