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? Even so, 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. Because of that, 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 That alone is useful..
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 Most people skip this — try not to..
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? Day to day, for cosine, it's the second term that flips. The signs flip depending on whether you're adding or subtracting. Also, for sine, the middle term changes from + to −. It's not as bad as it looks — once you work through a few examples, the structure clicks Still holds up..
Why "Exact Value" Matters
When your calculator gives you sin(75°), you get a decimal approximation: about 0.Because of that, 9659. 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.
The sum and difference formulas give you exactly that. And instead of "approximately 0. Which means 966," you get (√6 + √2)/4. It's precise, it's exact, and it shows you understand what's actually happening under the hood Worth keeping that in mind..
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 The details matter here..
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 That alone is useful..
It also builds intuition. Now, 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°).
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.
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 Simple as that..
Finding Cosine Using the Difference Formula
Try cos(15°). That's 45° − 30° And that's really what it comes down to..
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.
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 The details matter here..
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 Surprisingly effective..
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 And that's really what it comes down to..
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 Turns out it matters..
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 It's one of those things that adds up..
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. 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. Because of that, 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. Day to day, 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.
It sounds simple, but the gap is usually here Simple, but easy to overlook..
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. You understand trig at a deeper level. And honestly, there's something satisfying about writing (√6 + √2)/4 instead of "about 0.966 Nothing fancy..
Start with the easy angles. Work through a few examples by hand. The pattern becomes second nature faster than you'd expect.
