Ratios & Identities: Transformation Formula - Product into Sum or Difference

Trigonometric identities are equations involving trigonometric functions that are true for every value of the variables where both sides of the equation are defined. Transformation formulas are a set of trigonometric identities that allow us to express products of trigonometric functions as sums or differences (and vice versa). These formulas can simplify the integration of trigonometric functions and solve trigonometric equations.

Transformation Formulas

The transformation formulas for converting products into sums or differences are derived from the sum and difference identities for sine and cosine. Here are the key formulas:

Product-to-Sum Formulas

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

Sum-to-Product Formulas

1. $\sin A + \sin B = 2 \sin\left(\frac{A + B}{2}\right) \cos\left(\frac{A - B}{2}\right)$
2. $\sin A - \sin B = 2 \cos\left(\frac{A + B}{2}\right) \sin\left(\frac{A - B}{2}\right)$
3. $\cos A + \cos B = 2 \cos\left(\frac{A + B}{2}\right) \cos\left(\frac{A - B}{2}\right)$
4. $\cos A - \cos B = -2 \sin\left(\frac{A + B}{2}\right) \sin\left(\frac{A - B}{2}\right)$

Important Points

Transformation Type	Formula	Use Case
Product-to-Sum	$\sin A \cdot \sin B$	Simplifying products of sine functions
Product-to-Sum	$\cos A \cdot \cos B$	Simplifying products of cosine functions
Product-to-Sum	$\sin A \cdot \cos B$	Simplifying products of sine and cosine functions
Product-to-Sum	$\cos A \cdot \sin B$	Simplifying products of cosine and sine functions
Sum-to-Product	$\sin A + \sin B$	Expressing sums of sine functions as a product
Sum-to-Product	$\sin A - \sin B$	Expressing differences of sine functions as a product
Sum-to-Product	$\cos A + \cos B$	Expressing sums of cosine functions as a product
Sum-to-Product	$\cos A - \cos B$	Expressing differences of cosine functions as a product

Examples

Example 1: Product-to-Sum

Express the product $\sin x \cdot \sin y$ as a sum.

Using the product-to-sum formula:

$$ \sin x \cdot \sin y = \frac{1}{2}[\cos(x - y) - \cos(x + y)] $$

Example 2: Sum-to-Product

Express the sum $\sin x + \sin y$ as a product.

Using the sum-to-product formula:

$$ \sin x + \sin y = 2 \sin\left(\frac{x + y}{2}\right) \cos\left(\frac{x - y}{2}\right) $$

Example 3: Product-to-Sum

Express the product $\cos x \cdot \cos y$ as a sum.

Using the product-to-sum formula:

$$ \cos x \cdot \cos y = \frac{1}{2}[\cos(x - y) + \cos(x + y)] $$

Example 4: Sum-to-Product

Express the difference $\cos x - \cos y$ as a product.

Using the sum-to-product formula:

$$ \cos x - \cos y = -2 \sin\left(\frac{x + y}{2}\right) \sin\left(\frac{x - y}{2}\right) $$

Conclusion

Transformation formulas are powerful tools in trigonometry. They allow us to convert between products and sums or differences of trigonometric functions, which can simplify complex expressions and make certain types of problems more manageable. Understanding and applying these formulas is essential for success in trigonometry, calculus, and other areas of mathematics that involve trigonometric functions.