Transformation Formulas

🔮 The Magic Toolbox of Trig Transformations

Turning complicated trig expressions into simple ones—like a wizard with a wand!

🎭 The Story: Your Shape-Shifting Toolbox

Imagine you have a magic toolbox. Inside are special tools that can change the shape of things without changing what they really are.

• A square of clay can become a flat pancake
• Two toys can merge into one
• One toy can split into two
• A tricky puzzle can turn into an easy one

That’s what Transformation Formulas do for trigonometry! They change how expressions look without changing what they mean.

📦 Tool #1: Power Reducing Formulas

What’s the Problem?

Sometimes we have things like sin²θ or cos²θ (sine or cosine squared). These are hard to work with in calculus and other math.

Solution: We can “reduce” the power from 2 down to 1!

🎯 The Magic Trick

Think of it like this:

• You have a big box (sin²θ)
• You want a flat box (something without the square)

The Formulas

sin²θ = (1 - cos2θ) / 2

cos²θ = (1 + cos2θ) / 2

tan²θ = (1 - cos2θ) / (1 + cos2θ)

🌟 Simple Example

Problem: Simplify sin²(30°)

Step 1: Use the formula

sin²(30°) = (1 - cos(60°)) / 2

Step 2: We know cos(60°) = 1/2

sin²(30°) = (1 - 1/2) / 2 = (1/2) / 2 = 1/4

Check: sin(30°) = 1/2, so sin²(30°) = (1/2)² = 1/4 ✓

Why This Matters

• Makes integration (calculus) much easier
• Turns squared terms into simple cosines
• The “2θ” means we use double the angle

📦 Tool #2: Product-to-Sum (Werner Formulas)

What’s the Problem?

Sometimes we multiply two trig functions together:

• sin(A) × cos(B)
• cos(A) × cos(B)
• sin(A) × sin(B)

Multiplication is messy! Sums are cleaner.

🎯 The Magic Trick

Imagine two friends holding hands (product). We can separate them into two friends standing apart (sum), but they’re still connected!

The Formulas

sinA × cosB = ½[sin(A+B) + sin(A-B)]

cosA × cosB = ½[cos(A+B) + cos(A-B)]

sinA × sinB = ½[cos(A-B) - cos(A+B)]

🌟 Simple Example

Problem: Convert sin(3x) × cos(x) to a sum

Step 1: Use the formula (A = 3x, B = x)

sin(3x) × cos(x) = ½[sin(3x+x) + sin(3x-x)]

Step 2: Simplify

= ½[sin(4x) + sin(2x)]

Result: A product became a sum! Much easier to integrate.

Memory Trick

• Sin × Cos → Both results are SIN
• Cos × Cos → Both results are COS
• Sin × Sin → Results are COS (but watch the minus!)

📦 Tool #3: Sum-to-Product Formulas

What’s the Problem?

The opposite situation! Sometimes we have:

• sin(A) + sin(B)
• cos(A) + cos(B)

And we need them as products instead.

🎯 The Magic Trick

It’s like combining two separate toys into one super toy!

The Formulas

sinA + sinB = 2 × sin((A+B)/2) × cos((A-B)/2)

sinA - sinB = 2 × cos((A+B)/2) × sin((A-B)/2)

cosA + cosB = 2 × cos((A+B)/2) × cos((A-B)/2)

cosA - cosB = -2 × sin((A+B)/2) × sin((A-B)/2)

🌟 Simple Example

Problem: Convert sin(5x) + sin(3x) to a product

Step 1: Find the averages

• A = 5x, B = 3x
• (A+B)/2 = (5x+3x)/2 = 4x
• (A-B)/2 = (5x-3x)/2 = x

Step 2: Apply the formula

sin(5x) + sin(3x) = 2 × sin(4x) × cos(x)

Result: A sum became a product!

The Pattern

Notice: We always use the average (A+B)/2 and the half-difference (A-B)/2.

📦 Tool #4: Weierstrass t-Substitution

What’s the Problem?

Some trig integrals are really hard. No formula seems to work.

🎯 The Magic Trick

The Big Idea: Replace ALL trig functions with one simple variable t.

It’s like having a universal adapter that works with ANY plug!

The Substitution

Let t = tan(θ/2) (tangent of half the angle)

Then magically:

sinθ = 2t / (1 + t²)

cosθ = (1 - t²) / (1 + t²)

tanθ = 2t / (1 - t²)

dθ = 2dt / (1 + t²)

🌟 Simple Example

Problem: Find sinθ when θ = 60°

Step 1: Find t

t = tan(30°) = 1/√3

Step 2: Use the formula

sin(60°) = 2t / (1 + t²)
         = 2(1/√3) / (1 + 1/3)
         = (2/√3) / (4/3)
         = (2/√3) × (3/4)
         = 6 / (4√3)
         = 3 / (2√3)
         = √3/2 ✓

Why It’s Powerful

• Turns ANY trig integral into an algebra problem
• Works when nothing else does
• Called “universal” because it always works

When to Use It

Use Weierstrass when you see integrals like:

• ∫ 1/(a + b·sinθ) dθ
• ∫ 1/(a + b·cosθ) dθ
• Any “nasty” trig integral

🗺️ The Complete Map

graph TD
    A["Transformation Formulas"] --> B["Power Reducing"]
    A --> C["Product-to-Sum"]
    A --> D["Sum-to-Product"]
    A --> E["Weierstrass"]

    B --> B1["sin²θ → ½-½cos2θ"]
    B --> B2["cos²θ → ½+½cos2θ"]

    C --> C1["sinA×cosB → sums"]
    C --> C2["Products → Sums"]

    D --> D1["sinA±sinB → products"]
    D --> D2["Sums → Products"]

    E --> E1["t = tan θ/2"]
    E --> E2["All trig → algebra"]

🎯 Quick Summary

💪 You’ve Got This!

These four tools are like superpowers for trigonometry:

1. Power Reducing squashes big powers into small ones
2. Product-to-Sum breaks apart multiplication
3. Sum-to-Product combines addition into neat packages
4. Weierstrass is your “emergency escape” that always works

Remember: Each tool transforms expressions into equivalent forms. The math stays the same—only the appearance changes!

Now you can tackle any trig transformation challenge! 🚀