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Calc 2 God - Polar Cheat Sheet

Calc 2 God - Polar Cheat Sheet









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.…