Product to Sum Formulae
And now, the Product to Sum Formulae:
sinAcosB = 1/2[sin(A + B) + sin(A - B)]
cosAsinB = 1/2[sin(A + B) - sin(A - B)]
cosAcosB = 1/2[cos(A - B) + cos(A + B)]
sinAsinB = 1/2[cos(A - B) - cos(A + B)]
These formulae can be verified using the Sum and Difference Formulae. Observe:
First, add the argument properties for sine:
sin(A + B) = sinA cosB + cosA sinB
sin(A - B) = sinA cosB - cosA sinB
sin(A+ B) + sin(A - B) = 2sinAcosB Now divide by 2.
Answer: 1/2[sin(A+ B) + sin(A - B)] = sinAcosB
Now, subtract the argument properties for sine:
sin(A + B) = sinA cosB + cosA sinB
sin(A - B) = sinA cosB - cosA sinB
sin(A+ B) - sin(A - B) = 2cosAsinB Now divide by 2.
Answer: 1/2[sin(A + B) - sin(A - B)] = cosAsinB
Add the argument properties for cosine:
cos(A + B) = cosA cosB - sinA sinB
cos(A - B) = cosA cosB + sinA sinB
cos(A + B) + cos(A - B) = 2cosAcosB Now divide by 2.
Answer: 1/2[cos(A + B) + cos(A - B)] = cosAcosB
Subtract the argument properties for cosine:
cos(A + B) = cosA cosB - sinA sinB
cos(A - B) = cosA cosB + sinA sinB
cos(A + B) - cos(A - B) = 2sinAsinB Now divide by 2.
Answer: 1/2[cos(A + B) - cos(A - B)] = sinAsinB
Example
Express sin13°cos48° as a sum.
Solution
Trig Frog suggests that you begin with the appropriate formula.
sinAcosB = 1/2[sin(A + B) + sin(A - B)]
sin13°cos48° = 1/2[sin(13° + 48°) + sin(13° - 48°)] Perform the operations and simplify.
= 1/2(sin61° + sin(-35°) Remember that sine is an odd function.
