Sum and Difference of Sines & Cosines
Big Idea: These formulas allow us to convert a sum or difference of two trigonometric functions into a product of trigonometric functions. This is incredibly useful for simplifying expressions, solving equations, and understanding wave interference.
Where Do These Formulas Come From?
Recall the product-to-sum identities (derived from addition formulas):
- $\sin A \cos B = \frac{1}{2}[\sin(A+B) + \sin(A-B)]$
- $\cos A \cos B = \frac{1}{2}[\cos(A-B) + \cos(A+B)]$
- $\sin A \sin B = \frac{1}{2}[\cos(A-B) - \cos(A+B)]$
By making the substitution $A + B = \alpha$ and $A - B = \beta$, which gives us $A = \frac{\alpha + \beta}{2}$ and $B = \frac{\alpha - \beta}{2}$, we derive the sum-to-product formulas below.