Honestly, trigonometry gets a bad rap. Most people hear "identities" and immediately think of dusty chalkboards and memorizing strings of letters that don’t seem to make any sense. But if you’re trying to understand how waves work—whether that’s the sound hitting your headphones or the light hitting your eyes—you’ve gotta deal with the trigonometry addition and subtraction formulas. They aren't just academic torture. They’re the secret sauce behind signal processing and even the way GPS satellites talk to your phone.
Think about it.
Math isn't just about finding x. It's about relationship building. Specifically, these formulas tell us how angles interact when they’re mashed together or pulled apart. If you know the sine of $30^{\circ}$ and the sine of $45^{\circ}$, you don't need a calculator to find the sine of $75^{\circ}$. You just need to know how they play together.
🔗 Read more: Seeing the Unseen: Why Images of Multicellular Organisms Still Blow Our Minds
The Core Formulas: The "Big Six"
Most textbooks present these as a wall of text. It's overwhelming. But let's look at what's actually happening. These formulas essentially allow us to expand the sine, cosine, or tangent of a sum or difference into terms of individual angles.
Let’s look at the sine sum and difference identities first:
$$\sin(A + B) = \sin A \cos B + \cos A \sin B$$
$$\sin(A - B) = \sin A \cos B - \cos A \sin B$$
Notice something? Sine is "friendly." It mixes itself with cosine, and it keeps the sign the same. If you’re adding angles, you’re adding the products.
Now, look at cosine. Cosine is a bit of a rebel.
$$\cos(A + B) = \cos A \cos B - \sin A \sin B$$
$$\cos(A - B) = \cos A \cos B + \sin A \sin B$$
Cosine stays with its own kind (cos with cos, sin with sin) and it flips the sign on you. You add the angles, you subtract the terms. It’s counterintuitive at first, but once you see the pattern, it sticks.
Where Does This Actually Come From?
We aren't just pulling these out of thin air. You can derive these using the distance formula on a unit circle, or if you’re feeling fancy, Euler’s formula: $e^{ix} = \cos x + i \sin x$.
Imagine two points on a unit circle at angles $A$ and $B$. The distance between them can be calculated using the coordinates $(\cos A, \sin A)$ and $(\cos B, \sin B)$. When you expand that distance formula and simplify it using the Pythagorean identity $\sin^2\theta + \cos^2\theta = 1$, these formulas just... appear. It’s like a magic trick that actually has a logical explanation.
Why Tangent Is Different
The tangent formulas look like a mess. I get it.
$$\tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}$$
$$\tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B}$$
It’s a fraction. It’s clunky. But remember that $\tan \theta$ is just $\frac{\sin \theta}{\cos \theta}$. If you divide the sine sum formula by the cosine sum formula and do a little algebraic gymnastics (dividing everything by $\cos A \cos B$), you end up right here. It’s all interconnected. None of these formulas exist in a vacuum.
Trigonometry Addition and Subtraction Formulas in the Real World
You might be wondering why anyone would bother doing this by hand in 2026.
Computers are fast, sure. But software engineers at companies like NVIDIA or researchers at NASA don't just "plug it in." They use these identities to simplify complex expressions before the computer ever sees them. This saves processing power. In 3D engine development, rotating an object in space requires shifting angles. Using an addition formula is often computationally cheaper than running a full coordinate transformation from scratch.
Radio Waves and Interference
Ever heard of "noise-canceling" headphones? They work because of wave interference. When two sound waves meet, their pressures add up. If you have two waves with slightly different frequencies or phases, you can use the product-to-sum identities (which are just the trigonometry addition and subtraction formulas rearranged) to figure out exactly how they will cancel each other out.
It’s basically silence through math.
Common Pitfalls (And How to Avoid Them)
The biggest mistake? Treating $\sin(A + B)$ like it's a distributive property problem.
$\sin(A + B)$ is NOT $\sin A + \sin B$. If you do that on a test or in a physics simulation, everything breaks. Let’s prove it with simple numbers. If $A = 30^{\circ}$ and $B = 30^{\circ}$:
- $\sin(30^{\circ} + 30^{\circ}) = \sin(60^{\circ}) = \frac{\sqrt{3}}{2} \approx 0.866$
- $\sin(30^{\circ}) + \sin(30^{\circ}) = 0.5 + 0.5 = 1.0$
They aren't the same. Not even close.
Another weird quirk is the cosine sign flip. It’s so easy to accidentally write a plus when it should be a minus. I always tell people to remember: "Cosine is Contrary." It does the opposite of what the sign in the parentheses says.
Practical Applications for Students and Pros
If you're staring at a problem involving an angle like $15^{\circ}$ or $105^{\circ}$, don't reach for the calculator yet.
👉 See also: Your Account Cannot Be Created at This Time: Why iCloud is Blocking You
- $15^{\circ} = 45^{\circ} - 30^{\circ}$
- $105^{\circ} = 60^{\circ} + 45^{\circ}$
By using the exact values from the unit circle (like $\frac{\sqrt{2}}{2}$ and $\frac{1}{2}$), you get an exact answer. In high-level engineering and physics, "exact" is always better than "rounded decimal." Decimal errors compound. If you're building a bridge or a flight path, those tiny rounding errors can turn into a catastrophe over long distances.
The Role of Symmetry
One thing that doesn't get talked about enough is how these formulas prove the symmetry of functions. For example, if you use the subtraction formula with $90^{\circ}$, you can prove that $\sin(90^{\circ} - \theta) = \cos \theta$. This confirms that sine and cosine are just shifted versions of each other—hence the name "co-sine" (the sine of the complement).
Taking the Next Steps
Mastering these isn't about memorization. It’s about pattern recognition.
To really get comfortable with the trigonometry addition and subtraction formulas, you should try deriving the double-angle formulas next. If you let $A = B$ in the sum formulas, you suddenly get $\sin(2A) = 2 \sin A \cos A$. It's a "level up" in your math journey that makes calculus way easier later on.
Start with these three actions:
- Sketch the derivation: Don't just look at the formulas. Draw a unit circle and try to visualize why $\cos(A-B)$ relates to the distance between two points. Physical drawing creates mental anchors.
- Verify with known angles: Pick $90^{\circ}$ and $0^{\circ}$. Plug them into the formulas and see if they spit out the values you already know. It’s a great way to build trust in the math.
- Apply to non-standard angles: Try to find the exact value of $\tan(75^{\circ})$ using $45^{\circ}$ and $30^{\circ}$. It's a classic exercise that forces you to handle complex fractions and radicals simultaneously.
These identities are the building blocks for almost everything in periodic motion. Once you see them as tools rather than chores, the whole "scary" side of trig starts to disappear.