The angle between two polar curves is defined as the angle of intersection at a point where the curves meet. For polar curves and , this angle is determined by the tangents to the curves at the point of intersection.
If the polar equations of the curves are and , and the curves intersect at , then the angle between their tangents is given by:
\cos \phi = \frac{r_1'(\theta_0)r_2'(\theta_0) + r_1(\theta_0)r_2(\theta_0)}{\sqrt{\left(r_1'(\theta_0)^2 + r_1(\theta_0)^2
ight)\left(r_2'(\theta_0)^2 + r_2(\theta_0)^2
ight)}}
Here:
and are the derivatives of and with respect to at .
The denominator accounts for the magnitude of the tangent vectors.
This formula is particularly useful in higher engineering mathematics to analyze the behavior of polar curves at their points of intersection.