Sum and Difference Formulas

🎭 The Magic of Combining Angles

Sum and Difference Formulas in Trigonometry

Imagine you’re a chef. You know how chocolate tastes. You know how strawberry tastes. But what happens when you mix them together? Something new!

Trigonometry works the same way. You know sin(30°). You know sin(45°). But what about sin(30° + 45°)? That’s where our magic mixing formulas come in!

🌟 Why Do We Need These Formulas?

Here’s the thing that trips people up:

sin(A + B) is NOT the same as sin(A) + sin(B)

It’s like saying mixing red and blue paint gives you… red plus blue? No! You get purple! Something completely different.

These formulas tell us exactly what happens when we combine angles.

🎪 The Six Magic Formulas

Think of these as your recipe cards for combining angles:

graph TD
    A["Sum & Difference Formulas"] --> B["SINE"]
    A --> C["COSINE"]
    A --> D["TANGENT"]
    B --> B1["sin A+B"]
    B --> B2["sin A-B"]
    C --> C1["cos A+B"]
    C --> C2["cos A-B"]
    D --> D1["tan A+B"]
    D --> D2["tan A-B"]

1️⃣ Sum Formula for Sine

The Formula:

sin(A + B) = sin(A)·cos(B) + cos(A)·sin(B)

Think of it like a dance! 🕺💃

When two dancers (A and B) come together:

• First dancer’s sine partners with second dancer’s cosine
• First dancer’s cosine partners with second dancer’s sine
• They ADD their moves together!

Easy Memory Trick:

“SINE-COS plus COS-SINE”

Say it out loud: “Sign a check, plus cause a sign!”

📝 Example:

Find sin(75°) by writing it as sin(45° + 30°)

Step 1: Break it down

• A = 45°, B = 30°

Step 2: Use our formula

sin(75°) = sin(45°)·cos(30°) + cos(45°)·sin(30°)

Step 3: Plug in values we know

= (√2/2)·(√3/2) + (√2/2)·(1/2)
= √6/4 + √2/4
= (√6 + √2)/4

Answer: sin(75°) = (√6 + √2)/4 ≈ 0.966

🎉 Magic! We found sin(75°) without a calculator!

2️⃣ Sum Formula for Cosine

The Formula:

cos(A + B) = cos(A)·cos(B) − sin(A)·sin(B)

The Cosine Clique 👯

Cosines like to hang out together, sines like to hang out together. But here’s the twist — when they meet, there’s drama (subtraction)!

Easy Memory Trick:

“COS-COS minus SINE-SINE”

“Cause a cause, minus sign a sign!”

⚠️ Watch out! Unlike sine’s formula, cosine uses SUBTRACTION!

📝 Example:

Find cos(75°) as cos(45° + 30°)

Step by step:

cos(75°) = cos(45°)·cos(30°) − sin(45°)·sin(30°)
         = (√2/2)·(√3/2) − (√2/2)·(1/2)
         = √6/4 − √2/4
         = (√6 − √2)/4

Answer: cos(75°) = (√6 − √2)/4 ≈ 0.259

3️⃣ Difference Formula for Sine

The Formula:

sin(A − B) = sin(A)·cos(B) − cos(A)·sin(B)

Almost the Same Dance! 💃

Remember the sum formula? This is its twin — but instead of adding moves, they subtract!

Easy Memory Trick:

Same pattern as addition, just flip the + to −

📝 Example:

Find sin(15°) as sin(45° − 30°)

sin(15°) = sin(45°)·cos(30°) − cos(45°)·sin(30°)
         = (√2/2)·(√3/2) − (√2/2)·(1/2)
         = √6/4 − √2/4
         = (√6 − √2)/4

Answer: sin(15°) = (√6 − √2)/4 ≈ 0.259

🤔 Notice something? sin(15°) = cos(75°)! That’s not a coincidence — they’re complementary angles!

4️⃣ Difference Formula for Cosine

The Formula:

cos(A − B) = cos(A)·cos(B) + sin(A)·sin(B)

The Plot Twist! 🎬

Here’s where cosine flips the script:

• Sum formula uses minus
• Difference formula uses plus

It’s like opposite day for cosine!

📝 Example:

Find cos(15°) as cos(45° − 30°)

cos(15°) = cos(45°)·cos(30°) + sin(45°)·sin(30°)
         = (√2/2)·(√3/2) + (√2/2)·(1/2)
         = √6/4 + √2/4
         = (√6 + √2)/4

Answer: cos(15°) = (√6 + √2)/4 ≈ 0.966

5️⃣ Sum Formula for Tangent

The Formula:

tan(A + B) = (tan(A) + tan(B)) / (1 − tan(A)·tan(B))

The Fraction Situation 📊

Tangent is sine divided by cosine, so its formula looks like a fraction. Think of it as:

• Top (numerator): Add the tangents
• Bottom (denominator): 1 minus their product

Visual Picture:

         tan(A) + tan(B)
tan(A+B) = ─────────────────
         1 − tan(A)·tan(B)

📝 Example:

Find tan(75°) as tan(45° + 30°)

We know: tan(45°) = 1, tan(30°) = √3/3

tan(75°) = (1 + √3/3) / (1 − 1·√3/3)
         = (3/3 + √3/3) / (3/3 − √3/3)
         = (3 + √3)/3 / (3 − √3)/3
         = (3 + √3) / (3 − √3)

Rationalize by multiplying by (3 + √3)/(3 + √3):

= (3 + √3)² / (9 − 3)
= (9 + 6√3 + 3) / 6
= (12 + 6√3) / 6
= 2 + √3

Answer: tan(75°) = 2 + √3 ≈ 3.732

6️⃣ Difference Formula for Tangent

The Formula:

tan(A − B) = (tan(A) − tan(B)) / (1 + tan(A)·tan(B))

The Flip Again! 🔄

Just like with sine and cosine:

• Sum: add on top, subtract on bottom
• Difference: subtract on top, add on bottom

Visual Picture:

         tan(A) − tan(B)
tan(A−B) = ─────────────────
         1 + tan(A)·tan(B)

📝 Example:

Find tan(15°) as tan(45° − 30°)

tan(15°) = (1 − √3/3) / (1 + 1·√3/3)
         = (3 − √3)/3 / (3 + √3)/3
         = (3 − √3) / (3 + √3)

Rationalize:

= (3 − √3)² / (9 − 3)
= (9 − 6√3 + 3) / 6
= (12 − 6√3) / 6
= 2 − √3

Answer: tan(15°) = 2 − √3 ≈ 0.268

🧠 The Pattern Cheat Code

Here’s the secret pattern to remember everything:

Memory Rhyme:

"Sine goes SAME, Cosine goes STRANGE, Tangent’s a fraction, both in range!"

🎯 Quick Reference: All Six Formulas

┌─────────────────────────────────────────────┐
│ SUM FORMULAS                                │
├─────────────────────────────────────────────┤
│ sin(A+B) = sinA·cosB + cosA·sinB           │
│ cos(A+B) = cosA·cosB − sinA·sinB           │
│ tan(A+B) = (tanA + tanB)/(1 − tanA·tanB)   │
└─────────────────────────────────────────────┘

┌─────────────────────────────────────────────┐
│ DIFFERENCE FORMULAS                         │
├─────────────────────────────────────────────┤
│ sin(A−B) = sinA·cosB − cosA·sinB           │
│ cos(A−B) = cosA·cosB + sinA·sinB           │
│ tan(A−B) = (tanA − tanB)/(1 + tanA·tanB)   │
└─────────────────────────────────────────────┘

🌈 Why This Matters

These formulas let you:

1. Find exact values for angles like 15°, 75°, 105° without a calculator
2. Simplify complex expressions in physics and engineering
3. Derive other identities (like double-angle and half-angle formulas!)
4. Solve real problems in waves, sound, and light

💪 You’ve Got This!

Remember our chef analogy? You now know:

• How to mix sine (same sign as the operation)
• How to mix cosine (opposite sign — the rebel!)
• How to mix tangent (the fraction formula)

Each formula is just a recipe. Practice using them, and soon they’ll feel as natural as knowing that red + blue = purple!

🎉 Congratulations! You’ve just mastered one of the most powerful tools in trigonometry!