Polar and Coordinate Geometry

🗺️ Coordinate Trigonometry: Your GPS to the Math Universe

Imagine you’re a treasure hunter with two different maps to the same island. One map uses “go 3 steps right, then 4 steps up” (that’s rectangular). The other says “walk 5 steps in the direction of 53°” (that’s polar). Both lead to the SAME treasure!

🧭 The Polar Coordinate System

What Is It?

Think of a lighthouse spinning around. To find any boat in the ocean, you need just TWO things:

1. How far away is the boat? (This is called r - the distance)
2. Which direction is the boat? (This is called θ - the angle)

That’s it! Every point in the world can be found with just distance + direction.

     90° (up)
       |
       |
180° ——•—— 0° (right)
    (origin)
       |
     270° (down)

Simple Example

A pirate says: “The treasure is at (5, 60°)”

This means:

• Walk 5 steps from where you stand
• In the direction of 60° (a bit up and to the right)

🎯 Key Insight: Polar = “How far?” + “Which way?”

🔄 Polar to Rectangular: Translating the Pirate Map

The Magic Formulas

When pirates give you polar coordinates but your phone uses rectangular (GPS-style), convert like this:

Why Does This Work?

Picture a right triangle:

• The slanted side is your distance r
• The bottom is your x (horizontal)
• The side going up is your y (vertical)

Cosine gives you the horizontal part. Sine gives you the vertical part.

Example: Decode the Treasure

Pirate says: (5, 53.13°)

Let’s convert:

• x = 5 × cos(53.13°) = 5 × 0.6 = 3
• y = 5 × sin(53.13°) = 5 × 0.8 = 4

📍 Answer: The treasure is at rectangular point (3, 4)!

🔄 Rectangular to Polar: Making Your Own Pirate Map

The Reverse Formulas

Now you found a point at (3, 4) and want to give polar directions:

Example: Create the Pirate Map

You’re at point (3, 4):

Step 1: Find distance r

• r = √(3² + 4²)
• r = √(9 + 16)
• r = √25 = 5

Step 2: Find angle θ

• θ = tan⁻¹(4/3)
• θ = tan⁻¹(1.333…)
• θ = 53.13°

📍 Answer: In polar, this is (5, 53.13°)

⚠️ Watch Out: When x is negative, add 180° to your angle!

📍 Plotting Polar Points

The 3-Step Dance

To plot a polar point like (4, 120°):

1. Start at the center (the origin)
2. Face the direction of 120° (point your finger up-and-left)
3. Walk 4 steps in that direction

What About Negative r?

Here’s a twist! (-3, 45°) means:

• Face 45° direction
• Walk backwards 3 steps
• Same as (3, 225°)!

graph TD
    A["Start at Origin"] --> B["Face angle θ"]
    B --> C{Is r positive?}
    C -->|Yes| D["Walk forward r steps"]
    C -->|No| E["Walk backward |r| steps"]
    D --> F["Mark your point!"]
    E --> F

Quick Plotting Examples

✨ Polar Form of Complex Numbers

The Magical Connection

Complex numbers like 3 + 4i have a secret polar identity!

Think of 3 + 4i as a point:

• 3 is the real part (horizontal)
• 4 is the imaginary part (vertical)

The Beautiful Formula

Any complex number can be written as:

z = r(cos θ + i sin θ) or z = r·cis(θ)

Even shorter: z = re^(iθ) (Euler’s form!)

Example: Transform 3 + 4i

We already know from earlier:

• r = √(3² + 4²) = 5
• θ = tan⁻¹(4/3) = 53.13°

So: 3 + 4i = 5(cos 53.13° + i sin 53.13°)

Or simply: 5 cis(53.13°)

🌟 Why This Rocks: Multiplying complex numbers? Just multiply the r’s and ADD the angles!

🎡 Rotation Matrix Using Trig

Spinning Points Around

Want to rotate a point around the origin? There’s a magic recipe!

The Rotation Matrix

To rotate a point (x, y) by angle θ:

| x' |   | cos θ  -sin θ | | x |
|    | = |               | |   |
| y' |   | sin θ   cos θ | | y |

Which gives us:

• x’ = x·cos(θ) - y·sin(θ)
• y’ = x·sin(θ) + y·cos(θ)

Example: Rotate 90° Counterclockwise

Rotate point (3, 0) by 90°:

• x’ = 3·cos(90°) - 0·sin(90°) = 3·0 - 0·1 = 0
• y’ = 3·sin(90°) + 0·cos(90°) = 3·1 + 0·0 = 3

📍 Result: (3, 0) rotates to (0, 3)

The point moved from the right to the top! ✨

graph TD
    A["Original #40;3,0#41;"] --> B["Apply rotation matrix"]
    B --> C["New point #40;0,3#41;"]

📐 Angle Between Two Lines

Finding Where Lines Meet

Two lines can be friends (parallel), enemies (perpendicular), or somewhere in between!

The Formula

If two lines have slopes m₁ and m₂:

tan(θ) = |m₁ - m₂| / (1 + m₁·m₂)

Example: What Angle?

Line 1 has slope m₁ = 2 Line 2 has slope m₂ = 0.5

tan(θ) = |2 - 0.5| / (1 + 2 × 0.5) tan(θ) = |1.5| / (1 + 1) tan(θ) = 1.5 / 2 = 0.75

θ = tan⁻¹(0.75) = 36.87°

Special Cases

🎢 Parametric Equations with Trig

Drawing with Time

Instead of y = f(x), we let BOTH x and y depend on a third variable t (like time):

• x = f(t)
• y = g(t)

The Famous Circle

A circle with radius r centered at origin:

• x = r·cos(t)
• y = r·sin(t)

As t goes from 0° to 360°, the point traces a perfect circle!

Example: Unit Circle Journey

When t = 0°: (cos 0°, sin 0°) = (1, 0) → Start on the right

When t = 90°: (cos 90°, sin 90°) = (0, 1) → Top of circle

When t = 180°: (cos 180°, sin 180°) = (-1, 0) → Left side

When t = 270°: (cos 270°, sin 270°) = (0, -1) → Bottom

Other Cool Parametric Curves

Ellipse (stretched circle):

• x = a·cos(t)
• y = b·sin(t)

Cycloid (wheel rolling):

• x = r(t - sin t)
• y = r(1 - cos t)

graph TD
    A["Choose parameter t"] --> B["Calculate x = f#40;t#41;"]
    A --> C["Calculate y = g#40;t#41;"]
    B --> D["Plot point #40;x,y#41;"]
    C --> D
    D --> E["Change t, repeat!"]
    E --> A

🎯 Quick Reference Summary

🌟 The Big Picture

You now have TWO LANGUAGES to describe any point:

• Rectangular (x, y): Like city street addresses
• Polar (r, θ): Like compass directions

Different problems prefer different languages. Converting between them is your superpower!

Remember our treasure hunt? Whether you say “(3, 4)” or “(5, 53.13°)”—you’ll find the same gold! 🏴‍☠️💰