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

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

Sum-to-product and product-to-sum formulas let you rewrite trig expressions in equivalent forms that are easier to work with. A sum like $\sin A + \sin B$ can become a product, and a product like $\cos A \cos B$ can become a sum. This flexibility is what makes these formulas so useful for simplifying expressions and solving trig equations.

Sum-to-Product Formulas

These formulas convert a sum or difference of two trig functions into a product. Each formula uses the average of the two angles, $\frac{A+B}{2}$, and the half-difference, $\frac{A-B}{2}$.

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

Notice the pattern: sine formulas pair a sine with a cosine, while cosine formulas pair two of the same function. The cosine difference formula has a negative sign out front, which is easy to forget.

Example: Rewrite $\sin 5x + \sin 3x$ as a product.

1. Identify $A = 5x$ and $B = 3x$.
2. Compute $\frac{A+B}{2} = \frac{5x+3x}{2} = 4x$ and $\frac{A-B}{2} = \frac{5x-3x}{2} = x$.
3. Apply the sine sum formula: $\sin 5x + \sin 3x = 2\sin(4x)\cos(x)$.

This product form is often much easier to factor or set equal to zero when solving equations.

Product-to-Sum Formulas

These formulas go the other direction: they convert a product of two trig functions into a sum or difference.

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

A quick way to remember: when both factors are cosines, the result involves cosines. When both are sines, the result also involves cosines but with a negative sign out front. Mixed pairs (sine × cosine) produce sines.

Example: Rewrite $\cos 3\theta \cos 5\theta$ as a sum.

1. Identify $A = 3\theta$ and $B = 5\theta$.
2. Apply the cosine-cosine formula: $\cos 3\theta \cos 5\theta = \frac{1}{2}[\cos(3\theta+5\theta) + \cos(3\theta-5\theta)]$.
3. Simplify: $= \frac{1}{2}[\cos 8\theta + \cos(-2\theta)]$.
4. Since $\cos(-x) = \cos x$, the final answer is $\frac{1}{2}[\cos 8\theta + \cos 2\theta]$.

Solving Equations with These Formulas

The real payoff comes when you use these formulas to solve trig equations. The general strategy is to convert the equation into a form where you can factor or isolate a single trig function.

Steps for solving:

1. Identify whether the equation contains a sum/difference or a product of trig functions.
2. Apply the appropriate formula to rewrite the expression.
3. Set each factor equal to zero (if you now have a product) or simplify further using other identities.
4. Solve each resulting equation for the variable.
5. Check your solutions in the original equation, since algebraic manipulation can sometimes introduce extraneous solutions.

Example: Solve $\sin 3x + \sin x = 0$.

1. Apply sum-to-product: $2\sin\!\left(\frac{3x+x}{2}\right)\cos\!\left(\frac{3x-x}{2}\right) = 0$, which gives $2\sin(2x)\cos(x) = 0$.
2. Set each factor to zero: $\sin(2x) = 0$ or $\cos(x) = 0$.
3. $\sin(2x) = 0$ when $2x = n\pi$, so $x = \frac{n\pi}{2}$. And $\cos(x) = 0$ when $x = \frac{\pi}{2} + n\pi$. The cosine solutions are already included in $x = \frac{n\pi}{2}$, so the full solution set is $x = \frac{n\pi}{2}$, where $n$ is any integer.

Related Trigonometric Formulas

Sum-to-product and product-to-sum formulas are actually derived from the angle addition and subtraction identities. You'll sometimes need to combine them with other identities to fully simplify an expression:

• Double-angle formulas express functions like $\sin 2A$ or $\cos 2A$ in terms of $\sin A$ and $\cos A$.
• Half-angle formulas express functions of $\frac{A}{2}$ in terms of $A$.

These come up together often. For instance, after applying a sum-to-product formula you might end up with $\sin(2x)$, which you can expand using the double-angle identity $\sin(2x) = 2\sin x \cos x$ if that helps you factor further.