Products to Sums and Differences

What Are Products to Sums and Differences?

The "products to sums and differences" formulas are trigonometric identities that transform the product of sine and cosine functions into a sum or difference. These formulas simplify complex expressions and make solving equations more manageable.

The Formulas:

1. $\sin A \cos B = \frac{1}{2}[\sin(A + B) + \sin(A - B)]$
2. $\cos A \sin B = \frac{1}{2}[\sin(A + B) - \sin(A - B)]$
3. $\cos A \cos B = \frac{1}{2}[\cos(A + B) + \cos(A - B)]$
4. $\sin A \sin B = -\frac{1}{2}[\cos(A + B) - \cos(A - B)]$ These formulas are essential for integration, signal processing, and simplifying expressions involving trigonometric functions.

How Are These Formulas Derived?

The products-to-sums formulas come from the addition and subtraction identities for sine and cosine:

• $\sin(A \pm B) = \sin A \cos B \pm \cos A \sin B$
• $\cos(A \pm B) = \cos A \cos B \mp \sin A \sin B$

Why Use These Formulas?

The products-to-sums identities help:

• Simplify trigonometric expressions.
• Solve trigonometric equations.
• Evaluate integrals involving trigonometric products.

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Worked Examples

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Summary:

• The products to sums and differences formulas convert products of sine and cosine functions into sums or differences.
• These are helpful for simplifying expressions and solving equations.
• Key formulas to remember:

$$\sin A \cos B = \frac{1}{2}[\sin(A + B) + \sin(A - B)]$$ $$\cos A \sin B = \frac{1}{2}[\sin(A + B) - \sin(A - B)]$$ $$\cos A \cos B = \frac{1}{2}[\cos(A + B) + \cos(A - B)]$$ $$\sin A \sin B = -\frac{1}{2}[\cos(A + B) - \cos(A - B)]$$

• Practice applying these formulas in different scenarios for mastery.