Product-to-Sum Calculator: Converting Trig Products into Sums
About the Author
Marko Šinko
Co-Founder & Lead Developer, AI Math Calculator
Try to integrate ∫ sin 5x cos 3x dx and you hit a wall immediately. There is no product rule for integration. Reverse chain rule needs the derivative of one factor sitting next to the other, and it isn’t there. Integration by parts sends you round in a circle back to where you started. A product-to-sum calculator gets you out in one move by rewriting that product as ½[sin 8x + sin 2x] — two terms you can integrate on sight. The four identities behind that step are the workhorses of trig integration, Fourier analysis, and every radio receiver built since 1918, and this page walks through when to reach for them and what goes wrong when you don’t.
Why a Product of Sines Blocks Every Standard Technique
Take ∫ sin 5x cos 3x dx and try the usual toolkit honestly. Substitution wants u = 5x, but then cos 3x becomes cos(3u/5) and the integral is no simpler. Parts gives you ∫ u dv with u = sin 5x and dv = cos 3x dx, which produces another product of a sine and a cosine — run it twice and you get the original integral back with a coefficient, an algebra trick that works for ex sin x but not here. The reason nothing lands is structural: differentiation and integration are linear, so they cope with sums beautifully and with products not at all.
So change the structure. Convert the product to a sum, integrate the pieces, done:
∫ sin 5x cos 3x dx
= ∫ ½[sin 8x + sin 2x] dx
= ½[−cos 8x / 8 − cos 2x / 2] + C
= −cos 8x / 16 − cos 2x / 4 + C
Three lines. The whole difficulty was in the first move, and it is mechanical once you know the four formulas. Notice the direction of travel here: this is the mirror image of what a sum-to-product calculator does. That page turns sums into products so you can use the zero-product property to solve equations. This one runs the conversion the other way, because integration and frequency analysis want sums. Same four identities, opposite goals.
The Four Formulas, and the Half That Trips People Up
| Product | Sum form | Symmetric in A, B? |
|---|---|---|
| sin A cos B | ½[sin(A+B) + sin(A−B)] | No — order matters |
| cos A sin B | ½[sin(A+B) − sin(A−B)] | No — order matters |
| cos A cos B | ½[cos(A+B) + cos(A−B)] | Yes |
| sin A sin B | ½[cos(A−B) − cos(A+B)] | Yes |
The two cosine-heavy rows are forgiving. Swap A and B in cos A cos B and nothing changes, because cos(A−B) = cos(B−A) — cosine is even. Same for sin A sin B. The mixed rows are not forgiving at all, and this is where most errors come from. sin 5x cos 3x expands to ½[sin 8x + sin 2x], while sin 3x cos 5x expands to ½[sin 8x + sin(−2x)] = ½[sin 8x − sin 2x]. Same two functions, same two angles, different answer. Sine is odd, so flipping the difference flips that term’s sign.
A quick sanity check catches this every time. Set A = B and see whether you recover something you already know. For sin A cos B that gives sin A cos A = ½[sin 2A + sin 0] = ½ sin 2A, which is the double angle identity rearranged. For sin A sin B it gives sin²A = ½[1 − cos 2A], the power-reduction formula that underpins every half angle formula. If your version of the identity doesn’t reproduce those, you’ve mixed up a sign.
Two Lines of Derivation Beat Four Lines of Memorisation
You don’t need to memorise the table. Write the two angle-addition formulas for cosine and stack them:
cos(A − B) = cos A cos B + sin A sin B
cos(A + B) = cos A cos B − sin A sin B
add: cos(A−B) + cos(A+B) = 2 cos A cos B
subtract: cos(A−B) − cos(A+B) = 2 sin A sin B
There are two of the four rows, and the factor of ½ appears for an obvious reason — you divide by the 2 that adding two copies of the same term created. Do the identical thing with the sine expansions, sin(A+B) = sin A cos B + cos A sin B and sin(A−B) = sin A cos B − cos A sin B, and adding gives sin A cos B while subtracting gives cos A sin B. Four rows, eight lines of work, no memory required. It is worth doing once by hand because it also explains the asymmetry: the sine expansions are the ones where the two terms are genuinely different, which is exactly why those rows care about order.
The Integral That Makes Fourier Series Work
Here is the application that matters most and gets explained least. Take any two whole-number frequencies m and n and integrate their product over a full period:
∫₀²ᵖⁱ sin(mx) sin(nx) dx = ∫₀²ᵖⁱ ½[cos((m−n)x) − cos((m+n)x)] dx
m ≠ n: both cosines complete whole cycles → 0
m = n: cos(0) = 1 survives → ½ · 2π = π
Read that second line carefully, because it is doing all the work. A cosine of a non-zero whole-number frequency integrated over its full period is exactly zero — equal area above and below the axis. So when m ≠ n, both terms die and the answer is 0. When m = n, the difference angle collapses to zero, cos(0) = 1 is a constant rather than a wave, and the integral returns π. Different frequencies are orthogonal; identical ones are not.
That single fact is why a Fourier series is computable. A function written as a sum of infinitely many sines and cosines looks hopeless — how do you isolate one coefficient out of infinitely many? You multiply the whole series by sin(nx) and integrate. Every cross term vanishes by orthogonality, one term survives, and the coefficient falls out alone. Without the product-to-sum identity there is no clean proof of that step, which makes these four formulas load-bearing for signal processing rather than decorative. Set the calculator above to sin A · sin B with A = 3x and B = 2x and it prints the zero; change B to 3x and it prints π. If you want to see where the coefficients then go, the Fourier series calculator picks the story up from there.
Two Frequencies In, Two Different Frequencies Out
Look at what cos A cos B = ½[cos(A+B) + cos(A−B)] says physically. Feed a 1,000 kHz wave and a 1,455 kHz wave into a multiplier and the output contains neither. What comes out is 2,455 kHz and 455 kHz — the sum and the difference. That is a mixer, and it is the foundation of the superheterodyne receiver that has been in essentially every radio since the 1920s. The incoming station gets multiplied down to a fixed 455 kHz intermediate frequency where one carefully built filter can do the selective work, instead of needing a tunable filter that stays sharp across the entire band.
This is worth contrasting with beats, because students routinely merge the two. Beats come from adding two nearby tones: 440 Hz plus 444 Hz gives one 442 Hz tone wavering 4 times a second, and the sum-to-product identity explains it. Mixing comes from multiplying: 440 Hz times 444 Hz gives genuine new tones at 884 Hz and 4 Hz. Adding preserves the original frequencies and modulates the amplitude; multiplying destroys them and manufactures two new ones. The frequency diagram in the calculator draws that second case explicitly — input spikes on the top axis, output spikes on the bottom.
The special case where the two inputs are identical is worth its own mention. Multiply a signal by a reference at its own frequency and the difference term lands at zero frequency: a DC level, proportional to how well the two match. That is a lock-in amplifier, and it is how researchers pull a signal buried a thousand times below the noise floor back out into daylight. It is the same mathematics as the m = n case in the orthogonality integral above, wearing lab clothes.
Squeezing Exact Values Out of Awkward Angles
The identities also handle exact-value questions where a decimal earns no marks. Evaluate sin 75° cos 15° without a calculator:
sin 75° cos 15° = ½[sin(75° + 15°) + sin(75° − 15°)]
= ½[sin 90° + sin 60°]
= ½[1 + √3/2] = (2 + √3) / 4 ≈ 0.933013
Neither 75° nor 15° is a value you know by heart, but their sum and difference are both unit circle angles. That is the pattern to hunt for: any pair whose sum and difference are both special angles will collapse this way. 105° and 45° give 150° and 60°. 22.5° and 67.5° give 90° and −45°. Once you start looking for the pairing, these questions stop being about trigonometry and start being about arithmetic.
One caution on ordering, since it bites here too. sin 15° cos 75° is not the same expression — it comes out as ½[sin 90° + sin(−60°)] = ½[1 − √3/2] = (2 − √3)/4 ≈ 0.066987. Both are legitimate answers to different questions. Check which function owns which angle before you write anything down.
Choosing a Direction: Product-to-Sum Calculator or Its Reverse
Converting is a tool, not an improvement in itself. If you just need a number, sin 5x cos 3x and ½[sin 8x + sin 2x] cost exactly the same to evaluate. Use this table to decide which way to run the identity:
| Your goal | Direction | Why |
|---|---|---|
| Integrate or differentiate | Product → sum | Both operations are linear; sums split, products don’t |
| Solve an equation equal to zero | Sum → product | A product is zero when any factor is zero |
| Find a Fourier coefficient | Product → sum | Orthogonality kills every cross term |
| Predict a mixer or modulator output | Product → sum | The sum form names the output frequencies directly |
| Explain beats between two tones | Sum → product | The product form exposes the amplitude envelope |
| Get an exact value from odd angles | Either — whichever lands on special angles | Test the sum and difference against the unit circle |
Five Ways This Goes Wrong
- Writing 2 instead of ½. Sum-to-product identities have a leading 2; product-to-sum identities have a leading ½. Learn them in the same sitting and the coefficients swap in your memory. Quick check: sin A cos A must give ½ sin 2A, not 2 sin 2A.
- Treating sin A cos B as symmetric. It isn’t. sin 3x cos 5x and sin 5x cos 3x differ in the sign of the difference term because sin(−θ) = −sin θ.
- Losing the minus in sin A sin B. That row is ½[cos(A−B) − cos(A+B)], difference first and sum negative. Reversing it flips your entire answer, and the sanity check sin²A = ½[1 − cos 2A] catches it instantly.
- Forgetting to integrate the inside. ∫ sin 8x dx is −cos 8x / 8, not −cos 8x. Half the lost marks on this topic are here rather than in the identity itself.
- Integrating in degrees. ∫ sin(x°) dx picks up a factor of 180/π. Calculus is written in radians; convert before you integrate, and treat any degree symbol inside an integral as a warning sign.
When the Difference Angle Collapses
Two edge cases behave differently from everything above, and both are useful rather than pathological. When A = B the difference angle is zero, so the identity degenerates into a power-reduction formula: cos²A = ½[1 + cos 2A] and sin²A = ½[1 − cos 2A]. These are how you integrate a squared trig function at all — ∫ sin²x dx is impossible as written and trivial as ∫ ½[1 − cos 2x] dx = x/2 − sin 2x / 4 + C. If you have ever wondered why the average value of sin² over a period is exactly ½, that constant term is the answer, and it is also where the √2 in RMS voltage comes from.
The second case is A = −B, which sends the sum angle to zero instead. sin A cos B where B = −A gives ½[sin 0 + sin 2A] = ½ sin 2A, and the mixed identity quietly turns into the double-angle formula again. Neither case breaks the calculator — it detects the collapse, drops the dead term, and shows the simplified form directly. For the wider family of rearrangements these identities belong to, the list of trigonometric identities is the reference to keep open, and if you need to confirm that a rearrangement you’ve written is actually valid, a trig identity verifier settles it faster than checking by hand.
