LALO'S CHEAT SHEET

LEVEL 1: COORDINATE CONVERSION

Switching between coordinate systems. This is the foundation.

Cartesian (x, y) to Polar (r, θ)

• Find the radius `r`: \(r = \sqrt{x^2 + y^2}\)
• Find the angle `θ`: \(\tan\theta = \frac{y}{x}\)
• PITFALL: Always check the quadrant of (x, y) to find the correct angle. Your calculator's `arctan` might give you the wrong one!

Polar (r, θ) to Cartesian (x, y)

• Find `x`: \(x = r\cos\theta\)
• Find `y`: \(y = r\sin\theta\)
• PITFALL: A negative radius `r` means you move in the OPPOSITE direction of the angle.

LEVEL 2: EQUATION CONVERSION

The four key strategies for converting entire equations.

• Strategy 1: Direct Substitution. Use \(x = r\cos\theta\) and \(y = r\sin\theta\). Best for simple lines like `x = 3` or `y = 5`.
• Strategy 2: The "Multiply by r" Trick. If you see a lone `cos(θ)` or `sin(θ)`, multiply the whole equation by `r` to create \(r^2\) and \(r\cos\theta\) or \(r\sin\theta\).
  Example: `r = 6cos(θ)` becomes `r² = 6r cos(θ)`, which is `x² + y² = 6x`.
• Strategy 3: Clear the Fraction. If the equation is a fraction, multiply both sides by the denominator first.
  Example: `r = 30 / (6sinθ + 43cosθ)` becomes `r(6sinθ + 43cosθ) = 30`, which is `6y + 43x = 30`.
• Strategy 4: Use Trig Identities. Rewrite `tan(θ)`, `sec(θ)`, etc., in terms of `sin(θ)` and `cos(θ)` first.
  Example: `r = 5csc(θ)` becomes `r = 5/sin(θ)`, which is `r sin(θ) = 5` or `y = 5`.

FINAL BOSS: CALCULUS WITH POLAR

The most important formulas for the test.

Slope of a Tangent Line

This is the big one. You MUST memorize it.

$$\frac{dy}{dx} = \frac{\frac{dr}{d\theta}\sin\theta + r\cos\theta}{\frac{dr}{d\theta}\cos\theta - r\sin\theta}$$

Horizontal & Vertical Tangents

• Horizontal Tangent (slope = 0): Set the NUMERATOR to zero.
  \( \frac{dr}{d\theta}\sin\theta + r\cos\theta = 0 \)
• Vertical Tangent (slope = undefined): Set the DENOMINATOR to zero.
  \( \frac{dr}{d\theta}\cos\theta - r\sin\theta = 0 \)

Finding Key Points (Optimization)

• Highest/Lowest Point: Find the maximum/minimum of the `y` coordinate. Optimize the function \(y(\theta) = r\sin\theta\).
• Rightmost/Leftmost Point: Find the maximum/minimum of the `x` coordinate. Optimize the function \(x(\theta) = r\cos\theta\).
• Farthest from Origin: Find the maximum of `r` itself. This is NOT the same as the highest point!