Question

write the sum as a product: sin5x+sin3x

Answer 1

\(
\sin(A+B) = \sin(A)\cos(B) + \cos(A)\sin(B) \\
\sin(A-B) = \sin(A)\cos(B) - \cos(A)\sin(B) \\
\text{Add both sides: } \\
\sin(A+B) + \sin(A-B) = 2\sin(A)\cos(B) \\
\)

Set A + B = 5x and A - B = 3x. Solve for A and B and plug it into the equation on the last line above.

Answer 2

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)