Trig Series Expansions

🌊 The Magic of Infinite Waves: Taylor Series for Trigonometry

🎭 The Story Begins: Breaking Waves into Tiny Ripples

Imagine you’re at the beach, watching ocean waves. Each big wave is actually made of countless tiny ripples all working together. That’s exactly what Taylor series do with trigonometry! They break complex wave-like functions into simple pieces we can add up.

Think of it like building with LEGO blocks. Instead of one complicated curved piece, we use many straight pieces stacked cleverly to create curves!

🎯 What You’ll Discover

graph TD
    A["Taylor Series Magic"] --> B["sin x Series"]
    A --> C["cos x Series"]
    A --> D["tan x Series"]
    A --> E[Machin's Formula for π]
    A --> F["Gregory Series for arctan"]

📖 Part 1: The Taylor Series for sin x

The Big Idea

sin x describes a perfect wave. But what if we could rebuild this wave using only simple powers of x?

The Magic Formula

$\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + …$

🍕 Pizza Slice Analogy

Imagine cutting a pizza into infinite slices. Each slice gets smaller and smaller. When you add them all up, you get the whole pizza!

• First slice (x): The biggest piece - a rough guess
• Second slice (-x³/6): A small correction
• Third slice (+x⁵/120): An even tinier fix
• And so on…

🔢 Simple Example

Let’s find sin(0.5) using just 3 terms:

Step 1: x = 0.5 Step 2: Calculate:

• First term: 0.5 = 0.5
• Second term: -(0.5)³/6 = -0.125/6 ≈ -0.0208
• Third term: (0.5)⁵/120 = 0.03125/120 ≈ 0.00026

Step 3: Add them: 0.5 - 0.0208 + 0.00026 ≈ 0.4795

The real value? sin(0.5) ≈ 0.4794 ✨ Amazing accuracy!

🎨 Pattern to Remember

• Only odd powers: x¹, x³, x⁵, x⁷…
• Signs alternate: +, -, +, -, …
• Denominators are factorials: 1!, 3!, 5!, 7!..

📖 Part 2: The Taylor Series for cos x

The Big Idea

If sin x is a wave starting at zero, cos x is the same wave but shifted - it starts at 1!

The Magic Formula

$\cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + …$

🏔️ Mountain Analogy

Think of cos x as standing on top of a mountain (at height 1). The series tells you how to calculate your exact height as you walk along the curved path.

🔢 Simple Example

Let’s find cos(0.5) using 3 terms:

Step 1: x = 0.5 Step 2: Calculate:

• First term: 1
• Second term: -(0.5)²/2 = -0.25/2 = -0.125
• Third term: (0.5)⁴/24 = 0.0625/24 ≈ 0.0026

Step 3: Add them: 1 - 0.125 + 0.0026 ≈ 0.8776

The real value? cos(0.5) ≈ 0.8776 🎯 Perfect!

🎨 Pattern to Remember

• Only even powers: x⁰, x², x⁴, x⁶…
• Starts with 1 (which is x⁰)
• Signs alternate: +, -, +, -, …
• Denominators are factorials: 0!, 2!, 4!, 6!..

📖 Part 3: The Taylor Series for tan x

The Big Idea

tan x = sin x / cos x, but it has its own beautiful series too!

The Magic Formula

$\tan x = x + \frac{x^3}{3} + \frac{2x^5}{15} + \frac{17x^7}{315} + …$

🎢 Rollercoaster Analogy

Unlike sin and cos which are gentle waves, tan x is like a wild rollercoaster. It shoots up to infinity, then appears from the bottom again!

🔢 Simple Example

Let’s find tan(0.3) using 2 terms:

Step 1: x = 0.3 Step 2: Calculate:

• First term: 0.3
• Second term: (0.3)³/3 = 0.027/3 = 0.009

Step 3: Add them: 0.3 + 0.009 = 0.309

The real value? tan(0.3) ≈ 0.3093 👍 Close!

⚠️ Important Warning

The tan series only works when |x| < π/2 (about 1.57). Outside this range, tan x goes to infinity!

🎨 Pattern to Remember

• Only odd powers: x¹, x³, x⁵…
• All signs are positive (+, +, +…)
• Coefficients follow the Bernoulli numbers pattern

📖 Part 4: Machin’s Formula for π

The Big Idea

Here’s a magical trick: we can calculate π (3.14159…) using arctan!

The Magic Formula

$\frac{\pi}{4} = 4 \arctan\left(\frac{1}{5}\right) - \arctan\left(\frac{1}{239}\right)$

🎪 The Magic Show

John Machin discovered this in 1706. Before computers, people used this formula to calculate π to hundreds of decimal places BY HAND!

🔢 Why It Works

When you expand both arctangent terms using the Gregory series (coming next!), you get a formula where the fractions shrink super fast. This means fewer calculations for more accuracy!

🎯 Simple Verification

Using the Gregory series:

• arctan(1/5) ≈ 1/5 - 1/(3×125) + 1/(5×3125) ≈ 0.1974
• arctan(1/239) ≈ 1/239 ≈ 0.00418

Now: 4 × 0.1974 - 0.00418 ≈ 0.7854

And π/4 ≈ 0.7854 ✨ It works!

📖 Part 5: Gregory Series for arctan

The Big Idea

arctan x answers: “What angle has this tangent value?” The Gregory series expands this into simple terms!

The Magic Formula

$\arctan x = x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + …$

🗺️ Treasure Map Analogy

If tan x tells you “walk this far and turn this amount,” arctan x reads the map backwards: “Given where you ended up, what turn did you make?”

🔢 Simple Example

Let’s find arctan(0.5) using 3 terms:

Step 1: x = 0.5 Step 2: Calculate:

• First term: 0.5
• Second term: -(0.5)³/3 = -0.125/3 ≈ -0.0417
• Third term: (0.5)⁵/5 = 0.03125/5 = 0.00625

Step 3: Add them: 0.5 - 0.0417 + 0.00625 ≈ 0.4646

The real value? arctan(0.5) ≈ 0.4636 radians 🎯

🌟 Special Case: Finding π

When x = 1: $\arctan(1) = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + …$

Since arctan(1) = π/4, we get: $\frac{\pi}{4} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + …$

This is the famous Leibniz formula for π! (Though it converges very slowly)

🧠 How They All Connect

graph TD
    A["Taylor's Theorem] --> B[sin x series]
    A --> C[cos x series]
    B --> D[tan x = sin/cos]
    C --> D
    D --> E[arctan inverts tan]
    E --> F[Gregory Series]
    F --> G[Machin's Formula"]
    G --> H["Calculate π!"]

🎯 Quick Comparison Table

💡 Why This Matters

1. Calculators use these! Your phone calculates sin, cos, tan using Taylor series
2. Computer graphics - Video games use series for smooth animations
3. Engineering - Building bridges, designing circuits all need these
4. Space travel - NASA uses these formulas to navigate rockets!

🎉 You Did It!

You just learned how mathematicians “unwrap” curvy wave functions into simple polynomial pieces. Like a chef knowing the recipe for a complex dish, you now know the ingredients that make up sin, cos, tan, and arctan!

Remember:

“Every wave is just ripples added together. Every curve is just straight pieces cleverly stacked.”

Keep exploring, keep calculating, and most importantly - keep having fun with math! 🚀