7.4 Sum-to-Product and Product-to-Sum Formulas

Sum-to-Product and Product-to-Sum Formulas

These formulas let you convert between products of trig functions and sums (or differences) of trig functions. Why would you want to do that? Because sometimes a product like $\sin A \cos B$ is hard to work with, but the equivalent sum is much easier to simplify or solve. The reverse is also true: a sum like $\sin A + \sin B$ can be factored into a product, which is often the key to solving trig equations.

Both sets of formulas come directly from the sum and difference identities you already know. They're not new math; they're just rearrangements of those earlier identities.

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Sum-to-Product and Product-to-Sum Formulas

Products to sums conversion

The product-to-sum formulas turn a product of two trig functions into a sum or difference. Here are the four formulas:

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

Notice the pattern: sin·cos formulas produce sines on the right side, cos·cos produces cosines with a plus, and sin·sin produces cosines with a negative sign out front. Paying attention to these patterns helps you avoid mixing up the formulas on a test.

How to apply them:

1. Identify which type of product you have (sin·cos, cos·sin, cos·cos, or sin·sin).
2. Determine your A and B values. The order matters for sin·cos vs. cos·sin.
3. Plug A and B into the matching formula.
4. Simplify the result.

Example: Write $\sin(3x)\cos(x)$ as a sum.

Here $A = 3x$ and $B = x$, and you have sin·cos, so use the first formula:

$\sin(3x)\cos(x) = \frac{1}{2}[\sin(3x + x) + \sin(3x - x)] = \frac{1}{2}[\sin(4x) + \sin(2x)]$

Sums to products transformation

The sum-to-product formulas go the other direction: they take a sum or difference of trig functions and rewrite it as a product. These are especially useful for solving equations, because setting a product equal to zero lets you use the zero-product property.

• $\sin A + \sin B = 2\sin\!\left(\frac{A+B}{2}\right)\cos\!\left(\frac{A-B}{2}\right)$
• $\sin A - \sin B = 2\cos\!\left(\frac{A+B}{2}\right)\sin\!\left(\frac{A-B}{2}\right)$
• $\cos A + \cos B = 2\cos\!\left(\frac{A+B}{2}\right)\cos\!\left(\frac{A-B}{2}\right)$
• $\cos A - \cos B = -2\sin\!\left(\frac{A+B}{2}\right)\sin\!\left(\frac{A-B}{2}\right)$

The key step every time is computing $\frac{A+B}{2}$ and $\frac{A-B}{2}$. A common mistake is forgetting to divide by 2, so double-check that.

Example: Write $\cos(5\theta) - \cos(3\theta)$ as a product.

Here $A = 5\theta$ and $B = 3\theta$. Compute the two pieces:

$\frac{A+B}{2} = \frac{5\theta + 3\theta}{2} = 4\theta \qquad \frac{A-B}{2} = \frac{5\theta - 3\theta}{2} = \theta$

Apply the cos - cos formula:

$\cos(5\theta) - \cos(3\theta) = -2\sin(4\theta)\sin(\theta)$

Applications of trigonometric formulas

Simplifying expressions: When you see a complicated trig expression, look for products or sums that match one of these formulas. Convert it, then combine with other identities you know (Pythagorean identity $\sin^2 x + \cos^2 x = 1$, double-angle formulas, etc.) to keep simplifying.

Evaluating exact values: You can use these formulas to find exact values of expressions that don't correspond to standard angles on their own.

For example, to evaluate $\cos(75°)\cos(15°)$:

1. Use the cos·cos product-to-sum formula with $A = 75°$ and $B = 15°$.
2. $\cos(75°)\cos(15°) = \frac{1}{2}[\cos(90°) + \cos(60°)]$
3. $= \frac{1}{2}[0 + \frac{1}{2}] = \frac{1}{4}$

Solving equations: Converting sums to products is a powerful technique for trig equations because you can factor.

For instance, to solve $\sin(3x) + \sin(x) = 0$:

1. Apply sum-to-product: $2\sin(2x)\cos(x) = 0$
2. Set each factor to zero: $\sin(2x) = 0$ or $\cos(x) = 0$
3. Solve each equation separately to find all solutions.

This factoring approach is often much faster than trying to expand everything with other identities.

Advanced Trigonometric Concepts

• These formulas work in both degrees and radians. In pre-calc and beyond, radians are the standard, so practice using radian inputs.
• All trig functions are periodic, which means equations typically have infinitely many solutions. When solving, find the solutions in one period first, then generalize using the period (e.g., add $2\pi n$ for sine and cosine, or $\pi n$ for tangent).
• In more advanced courses, these formulas connect to Euler's formula ($e^{i\theta} = \cos\theta + i\sin\theta$), which provides an elegant way to derive all of these identities from exponential properties. You don't need Euler's formula for this course, but it's good to know the connection exists.