Sketching polar curves a spiral

Sketching Polar Curves - A Spiral

Polar curves are curves that are defined using polar coordinates rather than Cartesian coordinates.
A spiral polar curve or r = θ, often referred to as an Archimedean spiral, moves farther away from the origin as θ increases.

Steps to draw a spiral polar curve:

Identify the initial and terminal points on your polar graph by looking at the limits given for θ.
Mark these points on your graph, along with key points between these limits, such as when θ=90° (π/2 radians) or 180° (π radians).
Plot the points by converting θ to degrees or radians, and then determining the corresponding r value using the given polar equation.
Draw the curve smoothly through these points, making sure the curve always spirals away from the origin.

The spiral polar curve r = θ can be converted to Cartesian coordinates using the formulas x = r cos θ and y = r sin θ.
Convert r to θ using the polar equation of the spiral curve, resulting in x = θ cos θ and y = θ sin θ.

A spiral polar curve rotates about the origin, receding from the origin as it does so.
It creates an infinite number of “loops”, each further from the origin than the last.
Polar curves do not only represent curves in the plane; they can also represent a family of surfaces in space through rotation about a coordinate axis.

Given the spiral polar curve r = θ, we can sketch it by plotting these set points for θ = 0°, 90°, 180°, and 270° and connecting them in a smooth spiral away from the origin.
The corresponding r values for these points on the curve will be 0, π/2, π, and 3π/2 respectively.

Sketching a spiral polar curve involves understanding polar coordinates and knowing how to convert between polar and Cartesian coordinates.
The key to sketching is to plot key points and ensure that the drawn spiral continually moves away from the origin.
Understanding the properties of the curve is also crucial for effective sketching in the Core Pure section of further mathematics.