In order to measure the angle between two curves, we measure the angle between the tangents to the curves at that point. And that is obtained by the formula below:

tan θ =

where θ is the angle between the 2 curves, and m_{1} and m_{2} are slopes or gradients of the tangents to the curve at the point of intersection.

Let’s look at this through an example below:

Find the acute angle between the curves y = x^{2} and y = (x – 3)^{2}

The curve intersects at the point (where the two equations are equal). So

x^{2} = (x – 3)^{2}

x^{2} = x^{2} – 6x + 9

6x – 9 = 0

x = =

y = x^{2} = =

The intersection point is

The tangent to y = x^{2} at has a gradient of

= 2x = = 3 = m_{1}

The tangent to y = (x – 3)^{2} at has a gradient of

= 2(x – 3)^{1} = ^{ } = = -3 = m_{2}

tan θ =

=

=

=

=

= =

tan θ =

θ =

= 36° 52′ (to the nearest minute)