Polar Formulas

\[(r,\theta)\]

• r is directed distance

• θ is angle

Conversion

\(x^{2} + y^{2} = r^{2}\), \(x = r\cos\theta\), \(y = r\sin\theta\)

\(\cos\theta = \frac{x}{r}\), \(\sin\theta = \frac{y}{r}\), \(\tan\theta = \frac{y}{x}\)

Coordinates

Rectangular → Polar

\[(\sqrt{x^2+y^2},arctan(\frac{y}{x}))\]

Polar → Rectangular

\[(rcos\theta,rsin\theta)\]

Equation Types

Generally, equations with \(\sin\) are symmetric to the y-axis (except for lemniscates), while equations with \(\cos\) are symmetric to the x-axis.

Lines

\(r = a\sec\theta\) is the same as \(x = a\)(vertical line)

\(r = a\csc\theta\) is the same as \(y = a\)(horizontal line)

Circles

Standard Circle

\(r = a\)

• The same as \(x^{2} + y^{2} = a^2\)

• Center at the origin

• Radius of a

Sine Circle

\(r = a\sin\theta\) or \(r = - a\sin\theta\)

• Radius of \(\frac{a}{2}\)

• a dictates how far up or down the top/bottom of the circle is

Cosine Circle

\(r = a\cos\theta\) or \(r = -a\cos\theta\)

• Radius of \(\frac{a}{2}\)

• a dictates how far left or right the edge of the circle is

Limacons

\(r = a \pm b\sinθ\) or \(r = a \pm b\cosθ\)

If…

a = b → Cardioid ♥

a < b → Inner loop

a > b and \(\frac{a}{b} < 2\) → Dimpled circle

a > b and \(\frac{a}{b} \geq 2\) → Flat circle

Roses

\(r = a\ sin\ b\theta\) or \(r = a\ cos\ b\theta\)

• If b is odd there are b petals, but if b is even there are 2b petals.

• Each petal is a units long.
• The period of these graphs is always π.

Lemniscates

Each petal is of length \(a\).

\(r^{2} = a^{2}sin2\theta\)

• Symmetrical to \(\theta = \frac{\pi}{4}\) (the same as y = x)

\(r^{2} = a^{2}cos2\theta\)

• Symmetrical to x-axis

❓ How do I make the math bigger?

Right-click or long press any of the math, then click Math Settings → Scale All Math... Type a number such as 125% or 200%.