The name "equiangular spiral" suggests that some collection of angles related to this curve are all the same -- equal -- constant. In this part of the module, we first explore what's constant about this type of spiral. Then we examine the relationship between the constant and certain angles.
where r0 is the value of r at theta = 0 and k is a constant that incorporates both a multiple of pi and a growth constant. (Your formula may not look exactly like this, but it should be equivalent.) Explain why the ratio
is constant. That is, the rate of growth of r is proportional to r itself.
Now we consider how that constant proportional growth rate is related to a constant angle. In the following figure, we show a segment of polar curve in red, along with three angles associated with a given point on the curve:
We see from the figure that alpha = theta + beta, so beta = alpha - theta. Now use a formula from trigonometry to relate tan(beta) to the tangents of the other two angles:
The tangents in this formula are easy to relate to the (x,y) coordinates of the point on the curve:
Find derivatives of x and y with respect to theta, and then combine the results to find dy/dx in terms of theta. You may want to use your helper application for this.