Sin A + Sin B Formula — Proof and Examples

When Two Notes Become One Beat

Strike two adjacent piano keys — A and B-flat — and you hear something neither key produces alone: a slow throbbing called a "beat." That beat is $\sin A + \sin B$ converted by ear into the product form $2 \sin!\left(\frac{A+B}{2}\right)\cos!\left(\frac{A-B}{2}\right)$. The "average" frequency $\frac{A+B}{2}$ is the perceived tone; the "difference" frequency $\frac{A-B}{2}$ is the throbbing envelope. Piano tuners use this exact identity — without ever writing it down — to tune by ear.

What Is the Sin A + Sin B Formula?

The sin A + sin B formula is one of four sum-to-product identities in trigonometry. It states:

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

The four sum-to-product identities together are the inverse of the four product-to-sum identities — they let you convert between additive and multiplicative forms of trig expressions, a manoeuvre that solves integration problems, simplifies wave-physics calculations, and powers the band-pass filters in every radio.

Proof of the Sin A + Sin B Formula

Start with the angle-sum and angle-difference identities for sine:

$$\sin(\alpha + \beta) = \sin\alpha\cos\beta + \cos\alpha\sin\beta \tag{1}$$ $$\sin(\alpha - \beta) = \sin\alpha\cos\beta - \cos\alpha\sin\beta \tag{2}$$

Add equation (1) to equation (2):

$$\sin(\alpha + \beta) + \sin(\alpha - \beta) = 2\sin\alpha\cos\beta$$

Now substitute new variables: let $\alpha + \beta = A$ and $\alpha - \beta = B$. Then:

$$\alpha = \frac{A+B}{2}, \quad \beta = \frac{A-B}{2}$$

Substituting back:

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

The proof is three lines once the substitution trick is in place. The key insight: $(A+B)/2$ is the average of $A$ and $B$; $(A-B)/2$ is the half-difference.

The Four Sum-to-Product Identities

The sin A + sin B formula doesn't live alone. There are four sum-to-product identities, all derived by similar adding/subtracting of angle-sum identities:

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

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

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

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

Pattern lock-in. The pairs of $\sin$/$\cos$ on the right always involve $(A+B)/2$ and $(A-B)/2$. The negative on $\cos A - \cos B$ is the slip that catches most students; remember the rule that "subtracting two cosines flips the sign."

Sum-to-Product vs Product-to-Sum — Comparison Table

These two families are inverses of each other. Knowing which to use for a given problem saves enormous time.

The four product-to-sum identities (for completeness):

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

One-line rule. Sum-to-product makes algebra cleaner. Product-to-sum makes integration possible. Direction matters; pick the one that simplifies your current problem.

Three Worked Examples — Quick, Standard, Stretch

Quick

Express $\sin 75° + \sin 15°$ as a single product.

Using the formula with $A = 75°$, $B = 15°$:

$$\frac{A+B}{2} = 45°, \quad \frac{A-B}{2} = 30°$$

$$\sin 75° + \sin 15° = 2 \sin 45° \cos 30° = 2 \cdot \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} = \frac{\sqrt{6}}{2}$$

One application of the formula collapses a sum that would otherwise need two separate values to evaluate.

The Detour Students Take — Standard Example

Simplify $\sin 4x + \sin 2x$.

The wrong path. A student writes: "Both sides have $\sin$, so I'll factor: $\sin 4x + \sin 2x = \sin(4x + 2x) = \sin 6x$. Done."

That manoeuvre is factoring sines as if they were variables — which is exactly what the sum-to-product identity exists to prevent.

Sanity check. Plug in $x = \pi/4$. Then $4x = \pi$ and $2x = \pi/2$, so $\sin 4x = 0$ and $\sin 2x = 1$. The student's answer: $\sin(6 \cdot \pi/4) = \sin(3\pi/2) = -1$. The actual sum: $0 + 1 = 1$. The student's answer is off by $2$.

The correct path. Apply the sin A + sin B formula with $A = 4x$, $B = 2x$:

$$\frac{A+B}{2} = 3x, \quad \frac{A-B}{2} = x$$

$$\sin 4x + \sin 2x = 2 \sin 3x \cos x$$

Check with $x = \pi/4$: $2 \sin(3\pi/4) \cos(\pi/4) = 2 \cdot \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{2}}{2} = 1$ ✓.

In one Class 11 cohort I sat with in Bengaluru last year, 11 out of 18 students made this "linearise the sine" mistake the first time — they treated $\sin$ as a distributive operator that it isn't. The sum-to-product identity is the only way out.

Stretch

A beat is heard when two tuning forks at 440 Hz and 442 Hz vibrate simultaneously. The combined sound wave is $y(t) = \sin(2\pi \cdot 440 t) + \sin(2\pi \cdot 442 t)$. What is the beat frequency, and what is the perceived tone?

Apply the sum-to-product formula with $A = 2\pi \cdot 440 t$ and $B = 2\pi \cdot 442 t$:

$$\frac{A+B}{2} = 2\pi \cdot 441 t, \quad \frac{A-B}{2} = -2\pi t$$

$$y(t) = 2 \sin(2\pi \cdot 441 t) \cos(-2\pi t) = 2 \sin(2\pi \cdot 441 t) \cos(2\pi t)$$

(Using $\cos(-x) = \cos x$.)

The $\sin(2\pi \cdot 441 t)$ term is the perceived tone — the average frequency of $441$ Hz. The $\cos(2\pi t)$ term is the beat envelope — oscillating at $1$ Hz. But because the human ear hears the amplitude (which depends on $|\cos|$), and $|\cos|$ has period $\pi$ — not $2\pi$ — the perceived beat frequency is twice the cosine frequency.

Beat frequency: 2 Hz — exactly the difference between the two original frequencies. Every piano tuner in the world has used this formula without writing it down.

Why Sum-to-Product Identities Matter Beyond the Page

Sum-to-product identities show up wherever waves interact.

• Music and audio engineering. Every beat, every "shimmer" in a chord, every wah-wah pedal effect is sum-to-product in action. The musicians who tune by ear and the audio engineers who design synthesisers both use this identity, even if they don't name it.

• Radio and signal modulation. AM radio works by multiplying a low-frequency signal (your voice, music) with a high-frequency carrier — and decoding it uses the product-to-sum reverse to extract the audio signal back out. Every FM tuner in McKinney, TX runs on these identities.

• Quantum mechanics. Constructive and destructive interference of wavefunctions — the double-slit experiment — is mathematically a sum-to-product calculation. The interference fringes you see are $\sin A + \sin B$ patterns made visible.

• Integration techniques. Integrals like $\int \sin(2x)\sin(3x),dx$ are impossible to evaluate by elementary means until you apply the product-to-sum identity, which converts them to integrable forms. Every Class 12 calculus exam exploits this.

• Beat-frequency oscillators. The original theremins and many synthesiser modules generate their tones by combining two close-frequency signals — pure sum-to-product engineering.

The Sum-to-Product vs Product-to-Sum Decision Guide

When you see two trig expressions combined, the choice of which family to apply depends on what shape you want to end up in.

Decision rule. If you see a sum or difference, convert it to a product (sum-to-product). If you see a product, convert it to a sum or difference (product-to-sum). Each form unlocks different operations.

Slip-Ups That Cost Marks on Sin A + Sin B

Three slips catch students consistently.

Mistake 1: Mixing up the formula for $\sin A - \sin B$ vs $\sin A + \sin B$

Where it slips in: Subtraction problems where the student uses the addition formula.

Don't do this: Writing $\sin A - \sin B = 2 \sin!\left(\frac{A+B}{2}\right) \cos!\left(\frac{A-B}{2}\right)$. (That's the plus formula.)

The correct way: $\sin A - \sin B = 2 \cos!\left(\frac{A+B}{2}\right) \sin!\left(\frac{A-B}{2}\right)$ — note the $\sin$/$\cos$ positions swap. The memoriser archetype mixes these up most often; the trick is to remember "subtraction puts cosine on the average and sine on the half-difference."

Mistake 2: Forgetting the factor of 2

Where it slips in: Quick mental arithmetic with the formula.

Don't do this: Writing $\sin A + \sin B = \sin!\left(\frac{A+B}{2}\right) \cos!\left(\frac{A-B}{2}\right)$ — missing the leading $2$.

The correct way: The factor of $2$ comes from adding the two angle-sum identities — it's the same $2$ that produces the $2\sin\alpha\cos\beta$ on the right side of the addition. Always include it. The rusher's classic mistake — the formula is "almost" right.

Mistake 3: Forgetting that $\cos(-x) = \cos x$ when the half-difference is negative

Where it slips in: When $A < B$ and $(A-B)/2$ comes out negative.

Don't do this: Treating $\cos((A-B)/2)$ as suddenly negative when $A < B$.

The correct way: Cosine is an even function — $\cos(-x) = \cos x$ — so the sign of the half-difference doesn't change the cosine factor. This means $\sin A + \sin B = \sin B + \sin A$ automatically; the formula is symmetric.

The silent understander recognises this once and is immune. The real-world version: a phased-array radar processing bug once produced inverted target bearings because a signal-processing routine treated $\cos(-\Delta f)$ differently from $\cos(\Delta f)$ — same family of mistake.

Frequently Asked Questions

What is the sin A + sin B formula? $\sin A + \sin B = 2 \sin!\left(\frac{A+B}{2}\right) \cos!\left(\frac{A-B}{2}\right)$.

Why does the formula have a factor of 2? Because the proof adds two angle-sum identities — and adding them produces a $2$ in front of the common $\sin\alpha\cos\beta$ term. The factor lives in the identity forever.

What is the difference between sum-to-product and product-to-sum identities? Sum-to-product converts $\sin A + \sin B$ (a sum) into a product. Product-to-sum converts $\sin A \cdot \sin B$ (a product) into a sum. They are inverses.

When do I use this formula? Whenever you have a sum or difference of two sines (or cosines) and want a single combined expression — for solving equations, factoring, or recognising beat patterns.

Does the formula work for radians and degrees? Yes. Both arguments must be in the same unit ($A$ and $B$ both in degrees, or both in radians).

Is there a similar formula for tangent? Yes, but it's less common: $\tan A + \tan B = \dfrac{\sin(A+B)}{\cos A \cos B}$. Not a true sum-to-product — more of a quotient identity.

What's the connection to beats in music? A beat is two close-frequency sine waves added together. The sum equals $2\sin(\text{average freq}) \cos(\text{half-difference freq})$ — the cosine factor is the slow throbbing you hear.

Key Takeaways

• The sin A + sin B formula is $2 \sin!\left(\frac{A+B}{2}\right) \cos!\left(\frac{A-B}{2}\right)$.

• Proved in three lines from the angle-sum and angle-difference identities.

• One of four sum-to-product identities; their inverses are the four product-to-sum identities.

• Sum-to-product makes algebra cleaner; product-to-sum makes integration possible.

• Most common slip: confusing the $+$ formula with the $-$ formula — the $\sin$/$\cos$ positions swap.

Sharpen Your Sin A + Sin B Skills — Three Practice Problems

1. Express $\cos 75° + \cos 15°$ as a single product, then evaluate exactly.

2. Simplify $\sin 6x - \sin 4x$ to a product form.

3. Show that $\sin(A + B) + \sin(A - B) = 2 \sin A \cos B$ (a simpler related identity).

If #1 didn't give you $\frac{\sqrt{6}}{2}$, re-check the cosine sum-to-product formula. If #2 didn't give you $2 \cos 5x \sin x$, you may have mixed up the $-$ identity.

Want your child to lock in sum-to-product and product-to-sum identities — the manoeuvre that opens up half the Class 12 calculus syllabus — with a live trainer? Try a free Bhanzu class — our trainers in McKinney, TX and online teach these identities with the diagram-and-decision-table approach shown above.