polars.md

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$$(r,\theta)$$

• 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 $$\frac{a}{b} &lt; 2$$ → 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