Use Sum or Difference Formula to Find Exact Value
Ever stared at a problem like sin(75°) and wondered how anyone is supposed to know that off the top of their head? Here's the thing — you don't need to memorize every weird angle. You just need to break them into pieces you already know.
That's exactly what the sum and difference formulas let you do. On the flip side, they're the secret weapon for finding exact trigonometric values without a calculator. And once you see how they work, problems that looked impossible become almost satisfying.
What Are the Sum and Difference Formulas?
At their core, these formulas tell you how to find the sine, cosine, or tangent of an angle that's actually two angles added or subtracted together. Instead of trying to handle something like 75° directly, you recognize it's 45° + 30° — two angles you know perfectly well.
The formulas look like this:
Sine:
- sin(A + B) = sin A cos B + cos A sin B
- sin(A − B) = sin A cos B − cos A sin B
Cosine:
- cos(A + B) = cos A cos B − sin A sin B
- cos(A − B) = cos A cos B + sin A sin B
Tangent:
- tan(A + B) = (tan A + tan B) / (1 − tan A tan B)
- tan(A − B) = (tan A − tan B) / (1 + tan A tan B)
Notice the pattern? The signs flip depending on whether you're adding or subtracting. Here's the thing — for sine, the middle term changes from + to −. For cosine, it's the second term that flips. It's not as bad as it looks — once you work through a few examples, the structure clicks.
People argue about this. Here's where I land on it.
Why "Exact Value" Matters
When your calculator gives you sin(75°), you get a decimal approximation: about 0.9659. Now, that's fine for most practical work. But in math class, on the SAT, or in problems where you're proving identities, you often need the exact answer — the one with square roots in it Took long enough..
The sum and difference formulas give you exactly that. Instead of "approximately 0.966," you get (√6 + √2)/4. It's precise, it's exact, and it shows you understand what's actually happening under the hood Easy to understand, harder to ignore..
Why This Matters (And Where It Shows Up)
Here's the thing most people don't realize at first: these formulas aren't just a trick for solving one type of problem. They show up everywhere in trigonometry.
When you're simplifying expressions, solving equations, or proving identities, you'll constantly run into angles that aren't "nice" on their own — but are nice when you break them apart. The ability to recognize 15° as 45° − 30°, or 105° as 60° + 45°, transforms what you can actually solve.
It also builds intuition. You're not just punching buttons on a calculator; you're understanding the relationships between angles. That matters whether you're heading into calculus, physics, or any field where math shows up.
How to Use the Sum and Difference Formulas
Let's walk through this step by step with real examples.
Step 1: Break the Angle into Familiar Pieces
The key is recognizing which "nice" angles add or subtract to give you the angle you need. The most useful ones to memorize are:
- 30° (π/6)
- 45° (π/4)
- 60° (π/3)
From there, you can build most common angles:
- 15° = 45° − 30°
- 75° = 45° + 30°
- 105° = 60° + 45°
- 120° = 60° + 60°
Step 2: Apply the Correct Formula
Let's find the exact value of sin(75°) And that's really what it comes down to..
Step 1: Recognize that 75° = 45° + 30°.
Step 2: Use the sine sum formula: sin(A + B) = sin A cos B + cos A sin B Worth keeping that in mind..
So:
- sin(75°) = sin(45° + 30°)
- = sin(45°)cos(30°) + cos(45°)sin(30°)
Step 3: Plug in the values you know:
- sin(45°) = √2/2
- cos(30°) = √3/2
- cos(45°) = √2/2
- sin(30°) = 1/2
So:
- sin(75°) = (√2/2)(√3/2) + (√2/2)(1/2)
- = √6/4 + √2/4
- = (√6 + √2)/4
That's your exact value. No decimals, no approximations The details matter here..
Finding Cosine Using the Difference Formula
Try cos(15°). That's 45° − 30°.
Using the cosine difference formula: cos(A − B) = cos A cos B + sin A sin B The details matter here..
- cos(15°) = cos(45° − 30°)
- = cos(45°)cos(30°) + sin(45°)sin(30°)
- = (√2/2)(√3/2) + (√2/2)(1/2)
- = √6/4 + √2/4
- = (√6 + √2)/4
Interesting — cos(15°) gives you the same result as sin(75°). That's not a coincidence, but that's a topic for another day Easy to understand, harder to ignore..
Working with Tangent
Tangent formulas are a bit messier, but they follow the same logic. Let's find tan(π/12), which is tan(15°).
Since 15° = 45° − 30°, use the tangent difference formula: tan(A − B) = (tan A − tan B) / (1 + tan A tan B)
- tan(15°) = (tan 45° − tan 30°) / (1 + tan 45° tan 30°)
- = (1 − 1/3) / (1 + 1·1/3)
- = (2/3) / (4/3)
- = 2/4
- = 1/2
So tan(15°) = 1/2. Clean and exact And it works..
Using Radians? Same Process
If you're working in radians, nothing changes except the labels. π/12 = π/4 − π/6. The formulas work identically — you're just using π/4 and π/6 instead of 45° and 30°.
Common Mistakes People Make
Forgetting which sign goes where. The most frequent error is using the wrong sign in the formula. Remember: sine has the opposite sign pattern from cosine. When in doubt, write the formula out before you plug anything in Turns out it matters..
Choosing the wrong angle pair. Not every angle breaks down nicely. If you pick 50° and try to write it as 45° + 30°, you'll get 75°, not 50°. Pick your pair carefully.
Skipping the unit conversion. If your problem uses degrees but your reference angles are in radians (or vice versa), you'll get the wrong answer. Pick one system and stick with it Easy to understand, harder to ignore..
Messing up the square roots. This is where neatness matters. Write out each step clearly — it's easy to lose a √2 or drop a factor of 2 when you're doing mental math.
Practical Tips That Actually Help
Memorize the three key angles. If you know sin, cos, and tan for 30°, 45°, and 60°, you can build almost anything. That's really all you need Simple, but easy to overlook. That alone is useful..
Write the formula first, every time. Don't try to hold it in your head. Write down sin(A + B) = sin A cos B + cos A sin B, then substitute. It prevents almost every sign error.
Check your answer with a calculator. Not to get the decimal — just to verify you're in the right ballpark. If you got (√6 + √2)/4, that's about 0.9659. If your calculator says something totally different, you know to recheck Most people skip this — try not to..
Look for complementary angles. Remember that sin(θ) = cos(90° − θ). Sometimes finding the cosine version is easier, then you just flip it.
Frequently Asked Questions
How do I know which formula to use?
Look at your angle. If it's written as a sum (like 75° = 45° + 30°), use the sum formula. Think about it: if it's a difference (like 15° = 45° − 30°), use the difference formula. The function (sin, cos, or tan) tells you which family of formulas to grab.
What if the angle doesn't split into 30°, 45°, or 60°?
Some angles don't have clean exact values using these standard angles. That's okay — the sum and difference formulas still work, but you might end up with expressions involving multiple square roots that don't simplify nicely. In those cases, a decimal approximation might be the practical answer.
Can I use these formulas in reverse?
Absolutely. If you have an expression like sin A cos B + cos A sin B, you can collapse it into sin(A + B). This is incredibly useful when you're simplifying trigonometric expressions or proving identities.
Do these work with radians?
Yes, identically. The formulas don't care whether you write 45° or π/4 — the relationships are the same. Just make sure you're consistent with your units throughout a problem.
What's the difference between the sum and difference formulas?
The sum formulas (like sin(A + B)) have a plus sign in the middle term for sine and a minus sign in the middle term for cosine. The difference formulas flip those signs. It's a small change, but it matters enormously Turns out it matters..
The Bottom Line
The sum and difference formulas aren't about memorizing more stuff — they're about seeing angles differently. Once you train yourself to look at 75° and think "that's 45° plus 30°," a huge range of problems opens up.
You get exact answers instead of approximations. So you understand trig at a deeper level. And honestly, there's something satisfying about writing (√6 + √2)/4 instead of "about 0.966.
Start with the easy angles. Here's the thing — work through a few examples by hand. The pattern becomes second nature faster than you'd expect.
