Sketching curves the curve r = asin 2θ

Sketching Curves with the Equation r = asin 2θ

The equation r = asin 2θ is a common form of polar representation of ellipses or curves. It is defined by two variables: ‘r’ which is the distance from the origin (O), and ‘θ’ which is the angle from the positive x-axis.
‘a’ is the amplitude of the curve and ‘2θ’ suggests a repeated pattern, or loop, every π radians (or 180 degrees), rather than the usual 2π radians (or 360 degrees) in a full revolution. This indicates a form of doubling effect on the curve.
Having a positive ‘a’ results in a curve above the initial axis while having a negative ‘a’ will produce a curve below the initial axis.

Identify the range of θ: As it’s a form of ellipse, θ varies from 0 to 2π.
Calculate r for key points θ: Calculate ‘r’ for key values of ‘θ’. These commonly include θ = 0, π/2, π, 3π/2 and 2π.
Plot the points and sketch the curve: Plot the coordinated points (r, θ) for ‘r’ against ‘θ’ on the polar grid or Cartesian plane and join the dots to sketch the curve.

Remember, when ‘r’ is negative, the point plotted is actually on the opposite side of the origin from the direction θ implies.

The curve represented by the equation r = asin 2θ is symmetric about the initial line (line at θ = 0) as sine is an odd function.
It forms a cardioid like shape with a loop. The size of the loop depends on the amplitude ‘a’.
In half a rotation (π radians or 180 degrees), the curve makes a full loop. Then in the second half of the rotation, it retraces exactly the same loop. Hence, it actually completes two full loops in one full revolution.

Consider the example of r = 3sin 2θ. Plotting the key values and sketching the curve would give you two loops each extending 3 units from the origin, forming a sort of four-leaf clover shape.
Upon sketching, you can clearly see the symmetry about the initial line and the doubling of loops as implied by the ‘2θ’.