Product-to-Sum and Sum-to-Product Formulas, precalculus lesson

Play two notes that are almost the same pitch and the sound swells and fades a few times every second. Adding two waves has produced something that behaves like a product, a fast wave inside a slow one. The identities here say exactly how sums and products of sines and cosines turn into each other, and every one of them comes from the angle addition formulas you already know.

Key terms

Starting from angle addition

Definition For all real $A$ and $B$ the sum and difference formulas are

\begin{aligned} &\sin(A + B) \\[0.4em] &= \sin A \cos B + \cos A \sin B, \\[0.8em] &\sin(A - B) \\[0.4em] &= \sin A \cos B - \cos A \sin B, \\[0.8em] &\cos(A + B) \\[0.4em] &= \cos A \cos B - \sin A \sin B, \\[0.8em] &\cos(A - B) \\[0.4em] &= \cos A \cos B + \sin A \sin B. \end{aligned}

Adding or subtracting the two formulas for the same function cancels one product on the right and doubles the other. For all real $A$ and $B$ this gives

\begin{aligned} &\sin(A + B) + \sin(A - B) \\[0.4em] &= 2 \sin A \cos B, \\[0.8em] &\sin(A + B) - \sin(A - B) \\[0.4em] &= 2 \cos A \sin B, \\[0.8em] &\cos(A - B) + \cos(A + B) \\[0.4em] &= 2 \cos A \cos B, \\[0.8em] &\cos(A - B) - \cos(A + B) \\[0.4em] &= 2 \sin A \sin B. \end{aligned}

Every identity that follows is one of these four lines, divided by 2 or read with new names for the angles.

In plain words The sum formula and the difference formula for the same function differ in a single sign. Add them or subtract them and one of the two products drops out.

Example Subtracting the two cosine formulas gives the last line.

\begin{aligned} &\cos(A - B) - \cos(A + B) \\[0.4em] &= (\cos A \cos B + \sin A \sin B) \\[0.4em] &\quad - (\cos A \cos B - \sin A \sin B) \\[0.4em] &= 2 \sin A \sin B. \end{aligned}

With $A = 60^\circ$ and $B = 30^\circ$ the left side is $\cos 30^\circ - \cos 90^\circ = \frac{\sqrt{3}}{2}$ , and the right side is $2 \cdot \frac{\sqrt{3}}{2} \cdot \frac{1}{2} = \frac{\sqrt{3}}{2}$ as well.

Product-to-sum identities

Definition For all real $A$ and $B$ ,

\begin{aligned} &\sin A \cos B \\[0.4em] &= \tfrac{1}{2} \big[ \sin(A + B) + \sin(A - B) \big], \\[0.8em] &\cos A \sin B \\[0.4em] &= \tfrac{1}{2} \big[ \sin(A + B) - \sin(A - B) \big], \\[0.8em] &\cos A \cos B \\[0.4em] &= \tfrac{1}{2} \big[ \cos(A - B) + \cos(A + B) \big], \\[0.8em] &\sin A \sin B \\[0.4em] &= \tfrac{1}{2} \big[ \cos(A - B) - \cos(A + B) \big]. \end{aligned}

In plain words A product of two sines or cosines is half of a sum or difference of two new terms, one at the sum of the angles and one at their difference. Watch the last line, where the term with the sum of the angles carries the minus sign.

Example With $A = 75^\circ$ and $B = 15^\circ$ the first identity gives

\begin{aligned} &\sin 75^\circ \cos 15^\circ \\[0.4em] &= \frac{1}{2} \big[ \sin 90^\circ + \sin 60^\circ \big] \\[0.8em] &= \frac{1}{2} \left( 1 + \frac{\sqrt{3}}{2} \right) \\[0.8em] &= \frac{2 + \sqrt{3}}{4} \approx 0.9330. \end{aligned}

Two awkward angles turned into two famous ones.

Sum-to-product identities

Definition For all real $u$ and $v$ ,

\begin{aligned} &\sin u + \sin v \\[0.4em] &= 2 \sin \frac{u + v}{2} \cos \frac{u - v}{2}, \\[0.8em] &\sin u - \sin v \\[0.4em] &= 2 \cos \frac{u + v}{2} \sin \frac{u - v}{2}, \\[0.8em] &\cos u + \cos v \\[0.4em] &= 2 \cos \frac{u + v}{2} \cos \frac{u - v}{2}, \\[0.8em] &\cos u - \cos v \\[0.4em] &= -2 \sin \frac{u + v}{2} \sin \frac{u - v}{2}. \end{aligned}

In plain words A sum or difference of two sines, or of two cosines, is twice a product. One factor uses the average of the two angles and the other uses half their difference. The minus sign in the last line is the one people drop most often.

Example With $u = 75^\circ$ and $v = 15^\circ$ the average is $45^\circ$ and half the difference is $30^\circ$ , so

\begin{aligned} \sin 75^\circ + \sin 15^\circ &= 2 \sin 45^\circ \cos 30^\circ \\[0.4em] &= 2 \cdot \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} \\[0.8em] &= \frac{\sqrt{6}}{2} \approx 1.2247. \end{aligned}

Renaming the angles

Definition For any real $u$ and $v$ , the numbers

A = \frac{u + v}{2}, \qquad B = \frac{u - v}{2}

satisfy $A + B = u$ and $A - B = v$ . Substituting them into the four lines that come from angle addition turns each line into one of the sum-to-product identities. So the two families state the same four facts, read in opposite directions.

In plain words You do not need to memorize a second family from scratch. Call the two angles of a sum $u$ and $v$ , and the product side uses their average and half their difference.

Example Start from $\cos(A - B) - \cos(A + B) = 2 \sin A \sin B$ . With $A + B = u$ and $A - B = v$ it reads

\begin{aligned} &\cos v - \cos u \\[0.4em] &= 2 \sin \frac{u + v}{2} \sin \frac{u - v}{2}. \end{aligned}

Multiplying both sides by $-1$ gives the last sum-to-product identity. A check with $u = 90^\circ$ and $v = 30^\circ$ shows $\cos 90^\circ - \cos 30^\circ = -\frac{\sqrt{3}}{2}$ and $-2 \sin 60^\circ \sin 30^\circ = -\frac{\sqrt{3}}{2}$ .

Products of oscillations

Definition A function $\sin(\omega t)$ or $\cos(\omega t)$ with $\omega > 0$ is an oscillation with angular frequency $\omega$ , which completes $\frac{\omega}{2\pi}$ cycles per unit of time. For constants $a > b > 0$ and every real $t$ , the product-to-sum identities give

\begin{aligned} \sin(at) \cos(bt) &= \frac{1}{2} \Big[ \sin\big( (a + b)t \big) \\[0.4em] &\quad + \sin\big( (a - b)t \big) \Big], \end{aligned}

and similar formulas for $\cos(at)\cos(bt)$ and $\sin(at)\sin(bt)$ . So a product of two oscillations of amplitude $1$ is a sum or difference of two oscillations of amplitude $\frac{1}{2}$ , with angular frequencies $a + b$ and $a - b$ .

In plain words Multiplying two waves does not create a strange new shape. It creates two ordinary waves, one at the sum and one at the difference of the original frequencies.

Example With $a = 5$ and $b = 3$ ,

\sin 5t \cos 3t = \frac{1}{2} \big( \sin 8t + \sin 2t \big).

At $t = \frac{\pi}{4}$ the left side is $\left( -\frac{\sqrt{2}}{2} \right) \left( -\frac{\sqrt{2}}{2} \right) = \frac{1}{2}$ , and the right side is $\frac{1}{2}(\sin 2\pi + \sin \frac{\pi}{2}) = \frac{1}{2}$ .

Beats

Definition For angular frequencies $\omega_1 > \omega_2 > 0$ and every real $t$ , the sum-to-product identity gives

\begin{aligned} &\cos(\omega_1 t) + \cos(\omega_2 t) \\[0.4em] &= 2 \cos\left( \frac{\omega_1 - \omega_2}{2} \, t \right) \\[0.8em] &\quad \cdot \cos\left( \frac{\omega_1 + \omega_2}{2} \, t \right). \end{aligned}

When $\omega_1$ and $\omega_2$ are close, the second cosine oscillates quickly at the average angular frequency, and the first cosine changes slowly. The slow cosine, together with the $2$ in front, sets the size of the fast swings, and its absolute value is called the envelope. The envelope reaches its peak of $2$ once every $\frac{2\pi}{\omega_1 - \omega_2}$ units of time, so the loud moments, called beats, come

\frac{\omega_1 - \omega_2}{2\pi} = f_1 - f_2

times per unit of time, where $f_1 = \frac{\omega_1}{2\pi}$ and $f_2 = \frac{\omega_2}{2\pi}$ are the frequencies in cycles per unit of time.

In plain words Two nearly equal tones sound like one tone at the average pitch whose volume rises and falls. The number of swells per second is the difference of the two frequencies.

Example Tones of $440$ and $444$ hertz, with $t$ in seconds, are modeled by $\cos(880\pi t)$ and $\cos(888\pi t)$ . Their sum is

\begin{aligned} &\cos(888\pi t) + \cos(880\pi t) \\[0.4em] &= 2 \cos(4\pi t) \cos(884\pi t). \end{aligned}

The fast factor has frequency $442$ hertz, the average pitch. The envelope $\lvert 2\cos(4\pi t) \rvert$ peaks every $\frac{1}{4}$ second, so you hear $4$ beats per second, which is $444 - 440$ .

Check yourself

Write $2 \cos 5x \sin 2x$ as a sum or difference of sines.

Show the answer

Use the product-to-sum identity

\begin{aligned} &\cos A \sin B \\[0.4em] &= \frac{1}{2} \big[ \sin(A + B) - \sin(A - B) \big] \end{aligned}

with $A = 5x$ and $B = 2x$ , and multiply both sides by $2$ . Then

2 \cos 5x \sin 2x = \sin 7x - \sin 3x.

Show that the following holds for every $x$ at which the left side is defined.

\frac{\sin 3x + \sin x}{\cos 3x + \cos x} = \tan 2x

Show the answer

By the sum-to-product identities, with average $2x$ and half difference $x$ ,

\frac{\sin 3x + \sin x}{\cos 3x + \cos x} = \frac{2 \sin 2x \cos x}{2 \cos 2x \cos x}.

The left side is defined exactly when $\cos 2x \cos x \neq 0$ . For those $x$ the factor $2\cos x$ cancels, leaving $\frac{\sin 2x}{\cos 2x} = \tan 2x$ . At $x = \frac{\pi}{2}$ the right side equals $\tan \pi = 0$ , but the left side is undefined, so the identity holds only on the domain of the left side.

Two tones of $300$ and $306$ hertz sound together. Write $\cos(612\pi t) + \cos(600\pi t)$ as a product and find the number of beats per second.

Show the answer

The average of the angular frequencies is $606\pi$ and half their difference is $6\pi$ , so

\begin{aligned} &\cos(612\pi t) + \cos(600\pi t) \\[0.4em] &= 2 \cos(6\pi t) \cos(606\pi t). \end{aligned}

The envelope $\lvert 2\cos(6\pi t) \rvert$ peaks every $\frac{1}{6}$ second, so there are $6$ beats per second, which is $306 - 300$ .

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

Key terms

Starting from angle addition

Product-to-sum identities

Sum-to-product identities

Renaming the angles

Products of oscillations

Beats

Check yourself

How this lesson connects

Builds on

Where this leads in Calculus