The Trig Identity Everyone Misses: sin 3x cos x − cos 3x sin x
You're staring at a mess of sines and cosines, trying to figure out what sin 3x cos x − cos 3x sin x actually simplifies to. That's why maybe it's for a calculus problem. Maybe you're studying for an exam. Maybe you just stumbled across it and got curious.
Here's the thing — this expression has a beautifully simple answer. And once you see the pattern, you'll recognize it everywhere in trigonometry Simple, but easy to overlook..
What Is sin 3x cos x − cos 3x sin x?
At first glance, it looks complicated. You've got angles multiplied together, sines and cosines mixed up, and no obvious way to make sense of it. But this expression is actually a textbook example of a trigonometric identity — specifically, one of the angle subtraction formulas.
The expression sin 3x cos x − cos 3x sin x is asking you to find the difference between two products: the sine of three-x times the cosine of x, and the cosine of three-x times the sine of x.
The key insight? This matches the sine difference formula exactly:
sin(A − B) = sin A cos B − cos A sin B
If you let A = 3x and B = x, then:
sin(3x − x) = sin 3x cos x − cos 3x sin x
And 3x − x = 2x.
So the whole expression simplifies to sin(2x).
That's it. That's the answer. What looked like a tangled mess collapses into a single, clean term.
Related Identities Worth Knowing
Once you see this one, you'll notice its cousins everywhere. The cosine difference formula works similarly:
cos(A − B) = cos A cos B + sin A sin B
And the sum formulas go the other direction:
sin(A + B) = sin A cos B + cos A sin B cos(A + B) = cos A cos B − sin A sin B
These four formulas — the sum and difference identities for sine and cosine — are the backbone of almost every trig simplification you'll ever do Less friction, more output..
Why This Identity Matters
Here's why you should care about this beyond just passing a test That's the part that actually makes a difference..
It saves time. Instead of expanding sin 3x using the triple-angle formula (which is messy) and then trying to wrangle it with cos x, you recognize the pattern in about two seconds. The triple-angle formula for sin 3x is sin 3x = 3 sin x − 4 sin³x — and you'd need to multiply that by cos x, then subtract (cos 3x)(sin x), which itself expands to (4 cos³x − 3 cos x)(sin x). That's a disaster. The identity shortcut skips all of that.
It builds intuition. When you start seeing sin 3x cos x − cos 3x sin x as "sin of a difference," your brain stops seeing random symbols and starts seeing structure. That matters when problems get harder — and they will Easy to understand, harder to ignore. Surprisingly effective..
It's everywhere. This exact expression shows up in physics (wave interference), engineering (signal processing), and calculus (integrals involving trig functions). Recognizing it isn't just academic — it's practical.
How to Use This Identity
Let's walk through it step by step so it's crystal clear.
Step 1: Identify the pattern.
You have two terms. Each term is a product of a sine or cosine of one angle multiplied by a cosine or sine of another angle. Specifically:
- Term 1: sin(something) × cos(something else)
- Term 2: cos(something) × sin(something else)
If the first term has sin A cos B and the second has cos A sin B, you're looking at sin A cos B − cos A sin B Nothing fancy..
Step 2: Apply the formula.
The sine difference formula says: sin A cos B − cos A sin B = sin(A − B)
So you subtract the second angle from the first: sin(3x − x) = sin(2x)
Step 3: Simplify.
3x − x = 2x. Done.
A Quick Example
Let's say x = 15°. Then the original expression is:
sin(45°) cos(15°) − cos(45°) sin(15°)
Using the identity: = sin(45° − 15°) = sin(30°) = 0.5
You could verify this numerically — it works every time. Plus, that's the beauty of trig identities. They're not approximations; they're exact relationships.
Common Mistakes People Make
Trying to expand everything. The biggest mistake is trying to use the triple-angle and double-angle formulas to break down sin 3x and cos 3x separately. It's unnecessary and creates a mountain of work out of an anthill. Always check if you have a sum or difference pattern first Less friction, more output..
Forgetting the sign. Remember: sin A cos B − cos A sin B gives you sin(A − B). If the expression were sin A cos B + cos A sin B, that would be sin(A + B). The sign between the two terms matters. It's sin A cos B minus cos A sin B, so you're subtracting the angles.
Confusing sine and cosine formulas. The cosine difference formula looks similar but has different signs: cos A cos B + sin A sin B = cos(A − B)
Notice the plus signs. It's easy to mix these up when you're first learning. A good way to remember: sine subtraction has the subtraction sign in both places (sin A cos B − cos A sin B), and cosine subtraction has plus signs in both places.
Practical Tips for Trig Simplification
Memorize the four big ones. The sum and difference formulas for sine and cosine will show up constantly. Write them on a flashcard, put them on your wall, repeat them until they're automatic. They're:
- sin(A + B) = sin A cos B + cos A sin B
- sin(A − B) = sin A cos B − cos A sin B
- cos(A + B) = cos A cos B − sin A sin B
- cos(A − B) = cos A cos B + sin A sin B
Look for matching angles. In our expression, the angles were 3x and x. They weren't the same, but they were related. When you see two different angles in a trig expression, ask yourself: could this be a sum or difference in disguise?
Check your work with a simple value. If you're ever unsure whether you applied the identity correctly, plug in x = 30° or x = 45° — angles with clean sine and cosine values — and verify both sides give you the same number.
Don't overcomplicate it. If an expression looks like it fits a standard identity, it probably does. Trig problems are designed to have elegant solutions. When you find yourself doing pages of algebra, pause and look for the shortcut.
FAQ
What is sin 3x cos x − cos 3x sin x simplified?
It simplifies to sin(2x). This comes directly from the sine difference formula: sin(3x − x) = sin 2x Simple, but easy to overlook..
Is this the same as the double-angle formula?
Yes, essentially. That's why the double-angle formula for sine is sin(2x) = 2 sin x cos x. Our identity gives you sin(2x) in a different form — as a difference of products rather than twice a product. Both are valid ways to express sin(2x).
What about sin x cos 3x − cos x sin 3x?
This is the same expression with the terms reversed. And sin x cos 3x − cos x sin 3x = sin(x − 3x) = sin(−2x) = −sin(2x). The order of the angles matters — pay attention to which angle is being subtracted from which And that's really what it comes down to..
How do I remember these formulas?
A simple trick: the function on the left (sin or cos) stays the same, and the operation inside matches the operation between the terms. Cos A cos B + sin A sin B gives you cos(A − B). Sin A cos B − cos A sin B gives you sin(A − B). The plus/minus signs mirror each other.
Where is this used in real life?
This identity appears in any field dealing with waves or oscillations — physics, electrical engineering, acoustics, signal processing. When you're combining two wave functions with different frequencies, these sum and difference formulas describe what comes out the other side.
The truth is, trig identities like this one aren't just busywork. In real terms, they're tools that let you see complexity and collapse it into simplicity. Once you train your eye to spot sin A cos B − cos A sin B, you'll never approach it the same way again. It goes from being a problem to being a pattern — and that's when math starts clicking.
