Sum-to-product formulas

Definition

Sum-to-product formulas convert the sum or difference of trigonometric functions into a product of trigonometric functions. These formulas simplify the process of solving and analyzing trigonometric expressions.

5 Must Know Facts For Your Next Test

1. Sine and cosine sum-to-product formulas are derived from angle addition and subtraction identities.
2. The formula for $\sin A + \sin B$ is $2 \sin \left(\frac{A+B}{2}\right) \cos \left(\frac{A-B}{2}\right)$.
3. The formula for $\cos A + \cos B$ is $2 \cos \left(\frac{A+B}{2}\right) \cos \left(\frac{A-B}{2}\right)$.
4. The formula for $\sin A - \sin B$ is $2 \cos \left(\frac{A+B}{2}\right) \sin \left(\frac{A-B}{2}\right)$.
5. These formulas are useful for integrating trigonometric functions and simplifying complex trigonometric expressions.

Review Questions

• What is the sum-to-product formula for $\sin A + \sin B$?
• How can you express $\cos A - \cos B$ using a product of trigonometric functions?
• Explain how sum-to-product formulas simplify the integration of trigonometric functions.

Related terms

Product-to-Sum Formulas: Formulas that convert products of trigonometric functions into sums or differences, often used to simplify integration and solve equations.

Angle Addition Identities: Identities that express the sine, cosine, and tangent of a sum or difference of angles in terms of the sines, cosines, and tangents of the individual angles.

Trigonometric Simplification: The process of using identities to rewrite trigonometric expressions in simpler forms.