Lesson 9.4 : Sum-to-Product and Product-to-Sum Formulas

New Concept

This lesson introduces two powerful sets of identities: product-to-sum and sum-to-product formulas. You will learn to convert products of sine and cosine into sums, and sums into products, a key skill for simplifying complex trigonometric expressions.

What’s next

Next, you'll see how these formulas work with interactive examples. Then, you'll apply them yourself through a series of practice cards and challenge problems.

Property

We can derive the product-to-sum formula from the sum and difference identities for cosine. If we add the two equations, we get:

Then, we divide by 2 to isolate the product of cosines:

Given a product of cosines, to express it as a sum: write the formula, substitute the given angles into the formula, and simplify.

Examples

• To write $2 \cos(\frac{5x}{2}) \cos(\frac{x}{2})$ as a sum, we use the formula: $2 \cdot \frac{1}{2} [\cos(\frac{5x}{2} - \frac{x}{2}) + \cos(\frac{5x}{2} + \frac{x}{2})] = \cos(2x) + \cos(3x)$.

• Express $\cos(4\theta)\cos(2\theta)$ as a sum: $\frac{1}{2}[\cos(4\theta - 2\theta) + \cos(4\theta + 2\theta)] = \frac{1}{2}[\cos(2\theta) + \cos(6\theta)]$.

Property

By adding the sum and difference formulas for sine, we can derive the product-to-sum formula for sine and cosine:

Dividing by 2 isolates the product of sine and cosine:

Examples

• To express $\sin(3x)\cos(x)$ as a sum: $\frac{1}{2}[\sin(3x+x) + \sin(3x-x)] = \frac{1}{2}[\sin(4x) + \sin(2x)]$.

• Write the product $\sin(5\theta)\cos(3\theta)$ as a sum: $\frac{1}{2}[\sin(5\theta+3\theta) + \sin(5\theta-3\theta)] = \frac{1}{2}[\sin(8\theta) + \sin(2\theta)]$.

Property

The product-to-sum formulas transform products of sines and/or cosines into sums or differences. The formula for the product of sines is derived by subtracting the cosine identities.

The complete set of formulas is:

Examples

• Write $\sin(5x)\sin(2x)$ as a difference: $\frac{1}{2}[\cos(5x-2x) - \cos(5x+2x)] = \frac{1}{2}[\cos(3x) - \cos(7x)]$.

• Express $\cos(3\theta)\sin(5\theta)$ as a difference: $\frac{1}{2}[\sin(3\theta+5\theta) - \sin(3\theta-5\theta)] = \frac{1}{2}[\sin(8\theta) - \sin(-2\theta)] = \frac{1}{2}[\sin(8\theta) + \sin(2\theta)]$.

Property

The sum-to-product formulas express sums of sine or cosine as products. They are derived from the product-to-sum identities by letting $\alpha = \frac{u+v}{2}$ and $\beta = \frac{u-v}{2}$. The formulas are:

Examples

• Write $\sin(7x) + \sin(3x)$ as a product: $2\sin(\frac{7x+3x}{2})\cos(\frac{7x-3x}{2}) = 2\sin(5x)\cos(2x)$.

• Express $\cos(5\theta) - \cos(3\theta)$ as a product: $-2\sin(\frac{5\theta+3\theta}{2})\sin(\frac{5\theta-3\theta}{2}) = -2\sin(4\theta)\sin(\theta)$.