Product-to-sum and sum-to-product formulas
Sometimes a product like ${\sin 5x\cos 3x}$ is easier to work with as a sum. Other times a sum like ${\sin 7x+\sin 3x}$ is easier as a product. This page gives formulas for both directions. They are used in calculus and in the study of sound waves.
Product-to-sum formulas
Add the formulas for ${\sin(A+B)}$ and ${\sin(A-B)}$:
$\begin{align*}&\sin(A+B)+\sin(A-B)\\&=2\sin A\cos B\end{align*}$
The ${\cos A\sin B}$ terms cancel. Dividing by ${2}$ gives the first formula below. Adding or subtracting the cosine formulas in the same way gives the others:
$\begin{align*}&\sin A\cos B\\&=\dfrac{1}{2}\left[\sin(A+B)+\sin(A-B)\right]\end{align*}$
$\begin{align*}&\cos A\sin B\\&=\dfrac{1}{2}\left[\sin(A+B)-\sin(A-B)\right]\end{align*}$
$\begin{align*}&\cos A\cos B\\&=\dfrac{1}{2}\left[\cos(A+B)+\cos(A-B)\right]\end{align*}$
$\begin{align*}&\sin A\sin B\\&=\dfrac{1}{2}\left[\cos(A-B)-\cos(A+B)\right]\end{align*}$
Notice the last one: it starts with ${\cos(A-B)}$, not ${\cos(A+B)}$.
Example 1: Write ${\sin 5x\cos 3x}$ as a sum.
Solution:
Use the first formula with ${A=5x}$ and ${B=3x}$. Then ${A+B=8x}$ and ${A-B=2x}$:
$\begin{align*}&\sin 5x\cos 3x\\&=\dfrac{1}{2}(\sin 8x+\sin 2x)\end{align*}$
Example 2: Find the exact value of ${\cos 75^\circ\cos 15^\circ}$.
Solution:
Use the formula for ${\cos A\cos B}$ with ${A=75^\circ}$ and ${B=15^\circ}$:
$\begin{align*}&\cos 75^\circ\cos 15^\circ\\&=\dfrac{1}{2}(\cos 90^\circ+\cos 60^\circ)\\&=\dfrac{1}{2}\left(0+\dfrac{1}{2}\right)=\dfrac{1}{4}\end{align*}$
Sum-to-product formulas
Now go the other way. In the product-to-sum formulas, put ${x=A+B}$ and ${y=A-B}$. Then ${A=\dfrac{x+y}{2}}$ and ${B=\dfrac{x-y}{2}}$, and the formulas become
$\begin{align*}&\sin x+\sin y\\&=2\sin\dfrac{x+y}{2}\cos\dfrac{x-y}{2}\end{align*}$
$\begin{align*}&\sin x-\sin y\\&=2\cos\dfrac{x+y}{2}\sin\dfrac{x-y}{2}\end{align*}$
$\begin{align*}&\cos x+\cos y\\&=2\cos\dfrac{x+y}{2}\cos\dfrac{x-y}{2}\end{align*}$
$\begin{align*}&\cos x-\cos y\\&=-2\sin\dfrac{x+y}{2}\sin\dfrac{x-y}{2}\end{align*}$
In words: a sum becomes a product of a function of the average ${\dfrac{x+y}{2}}$ and a function of half the difference ${\dfrac{x-y}{2}}$.
Watch out: The last formula has a minus sign in front.
Example 3: Write ${\sin 7x+\sin 3x}$ as a product.
Solution:
The average is ${\dfrac{7x+3x}{2}=5x}$, and half the difference is ${\dfrac{7x-3x}{2}=2x}$. So
${\sin 7x+\sin 3x=2\sin 5x\cos 2x}$
Example 4: Find the exact value of ${\cos 105^\circ+\cos 15^\circ}$.
Solution:
The average is ${60^\circ}$, and half the difference is ${45^\circ}$:
$\begin{align*}&\cos 105^\circ+\cos 15^\circ\\&=2\cos 60^\circ\cos 45^\circ\\&=2\cdot\dfrac{1}{2}\cdot\dfrac{\sqrt{2}}{2}=\dfrac{\sqrt{2}}{2}\end{align*}$
Example 5: Prove the identity
$\dfrac{\sin 3x+\sin x}{\cos 3x+\cos x}=\tan 2x$
Solution:
Use the sum-to-product formulas on the top and on the bottom. In both, the average is ${2x}$ and half the difference is ${x}$:
$\begin{align*}&\dfrac{\sin 3x+\sin x}{\cos 3x+\cos x}\\&=\dfrac{2\sin 2x\cos x}{2\cos 2x\cos x}\\&=\dfrac{\sin 2x}{\cos 2x}=\tan 2x\end{align*}$
An application: beats
When two musical notes with nearly the same frequency are played together, you hear the sound get louder and softer in a steady rhythm. These pulses are called beats. Musicians use them to tune instruments.
The sum-to-product formula explains them. For a small example, add two waves with nearly the same speed:
${\cos 22t+\cos 18t=2\cos 20t\,\cos 2t}$
The product is a fast wave, ${\cos 20t}$, whose height is set by a slow wave, ${2\cos 2t}$. The slow wave makes the loudness swell and fade:
Example 6: Two tuning forks sound at ${442}$ Hz and ${438}$ Hz. The sound is
${y=\cos(884\pi t)+\cos(876\pi t)}$
Write it as a product, and find how many beats are heard each second.
Solution:
The average of ${884\pi t}$ and ${876\pi t}$ is ${880\pi t}$, and half the difference is ${4\pi t}$:
${y=2\cos(880\pi t)\cos(4\pi t)}$
The fast wave ${\cos(880\pi t)}$ has frequency ${\dfrac{880\pi}{2\pi}=440}$ Hz: we hear a note of ${440}$ Hz. Its loudness follows ${|2\cos(4\pi t)|}$. The function ${\cos 4\pi t}$ has period ${\dfrac{2\pi}{4\pi}=\dfrac{1}{2}}$ second, and its absolute value peaks twice in each period. So there are ${4}$ loud pulses per second.
That is the difference of the two frequencies: ${442-438=4}$ beats per second.
Summary
- Product to sum: for example, ${\sin A\cos B}$ is half of ${\sin(A+B)+\sin(A-B)}$.
- Sum to product: for example, ${\sin x+\sin y}$ is ${2}$ times the sine of the average of ${x}$ and ${y}$ times the cosine of half their difference.
- All of these come from adding or subtracting the sum and difference formulas.
- Two notes with frequencies ${f_1}$ and ${f_2}$ produce ${|f_1-f_2|}$ beats per second.