Pattern Brush? Why not use an Effect!
Pattern brush is a lot of work to create a sine wave. The pattern brush technique is also compounded by the complexity of generating a pattern brush suitable for wrapping around the radius of the circle. Otherwise the zero crossings (point that crosses the perimeter of the shape) will not be smooth. The elegance will be lost. Adobe Illustrator already has a tool for creating sinusoidal waves on arbitrary paths.
Step 1: Make a shape
Step 2: Apply Effect
Use the "Zig Zag" effect from the "Distort and Transform" submenu of the "Effect" menu.
The "Smooth" radio button will create a sinusoid.
Step 3: Rotate and Copy
First measure the angle between ridges. 90 degrees (top) - 70 degrees gives a 20 degree spacing. Half which is our rotation angle.
The effect can be expanded to create actual paths, or left as-is for flexibility. The zig zag tool is very flexible and can be used in a wide variety of situations.
This image shows a straight line with the zig zag effect applied three times. No other changes were made. This shape would have been difficult to create otherwise!