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Contents
Factor Formulae
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Factor Formulae
The
factor formulae are used to convert the product of two trigonometric functions
into a sum or difference, and vice versa. They are very useful, especially
when dealing with multiple angles such as sin5A or cos7A.
Product
Formulae
2
sinA cosB = sin (A + B) + sin (A - B)
2
cosA sinB = sin (A + B) - sin (A - B)
2
cosA cosB = cos (A + B) + cos (A - B)
2
sinA sinB = -cos (A + B) + cos (A - B)
These
formulae can be derived from applying addition formulae to the R.H.S. of the
identity:
sin (A + B) + sin (A - B)
= sinA cosB + cosA sinB + sinA cosB - cosA sinB
= 2 sinA cosB
The
rest can be proved similarly. Do it yourself as an exercise. It will reinforce
your understanding of addition formulae as well.
Sum
Formulae
 This
set of formulae are derived from the product formulae above, using the
substitutions X = A + B and Y = A - B. We shall use the first one to prove.
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