How to sketch a polar curve using a radius-versus-angle graph

A radius-versus-angle graph lets us use a familiar sine or cosine shape to organize a polar sketch. The horizontal axis records the angle, and the vertical axis records the radius. We then transfer those values into the polar plane in order.

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Choose angles that give useful radius values

For a sine or cosine of $n\theta$, increments of $\frac{\pi }{2n}$ make the trig argument change by $\pi /2$. That gives the simple trig values $0$, $1$, and $-1$. On this intermediate graph, a point is written $(\theta ,r)$; in polar coordinates, we write $(r,\theta )$.

Make a radius table for a circle

For $r=6\cos \theta$, use increments of $\pi /2$.

Radius values for six cosine theta

Angle	Radius
$0$	$6$
$\frac{\pi }{2}$	$0$
$\pi$	$-6$
$\frac{3\pi }{2}$	$0$
$2\pi$	$6$

On a radius-versus-angle graph, these points lie on a cosine wave. Its zeros tell us when the polar curve reaches the pole. Its negative values tell us to plot in the direction opposite the angle’s ray.

On this intermediate graph, θ is horizontal and r is vertical. The cosine wave organizes the values that we transfer to the polar plane.

Transfer the points to the polar plane

At $\theta =0$, we plot $(6,0)$. At $\theta =\pi /2$, the radius is zero. At $\theta =\pi$, the radius is $-6$, so we return to the point six units to the right of the pole. The intermediate angles trace the circle smoothly.

The polar graph is a circle with center (3, 0) and radius 3. It is traced once as θ runs from 0 to π.

Use smaller angle steps for a rose

For $r=2.5\cos \left(3\theta \right)$, solve $3\theta =\pi /2$ to get steps of $\pi /6$.

Beginning the three-petal rose

Angle	Radius
$0$	$2.5$
$\frac{\pi }{6}$	$0$
$\frac{\pi }{3}$	$-2.5$
$\frac{\pi }{2}$	$0$
$\frac{2\pi }{3}$	$2.5$

Transfer the points in angle order. The negative radius at $\theta =\pi /3$ places a petal tip in the opposite direction, at angle $4\pi /3$. Continuing the table traces all three petals.

Apply the same process to a cardioid

For $r=1.8-1.8\sin \theta$, the radii at $0$, $\pi /2$, $\pi$, $3\pi /2$, and $2\pi$ are $1.8$, $0$, $1.8$, $3.6$, and $1.8$. The zero gives the cusp at the pole, and the largest radius points downward. Transferring the values gives a downward-facing cardioid.

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