Angle of intersection

When the graphs intersect with two functions, the tangents form two angles with each other. The smaller angle is called the angle of intersection.


The picture: By cutting the two tangents (red) two angles are formed: $\gamma$ and $\gamma'$.
$\gamma$ is the angle of intersection.


!

Remember

$f$ and $g$ intersect at point $x$.

First, you need the slope angles of the functions:
$\alpha=\arctan(f'(x))$
$\beta=\arctan(g'(x))$

The angle of intersection has the smaller value:
$|\alpha-\beta|$ or
$180^\circ-|\alpha-\beta|$
i

Method

  1. Take derivatives
  2. Calculate slopes
  3. Calculate slope angle
  4. Specify the angle of intersection

Example

What is the angle of intersection of the functions $f(x)=x^2$ and $g(x)=x+2$ at the intersection $P(2|4)$.

  1. Take derivatives

    $f(x)=x^2$
    $f'(x)=2x$

    $g(x)=x+2$
    $g'(x)=1$

  2. Calculate slopes

    The slope of the two functions at the intersection is calculated ($x=2$).
    $f'(2)=2\cdot2=4$
    $g'(2)=1$

  3. Calculate slope angle

    $\alpha=\arctan(f'(x))$
    $\alpha=\arctan(4)\approx75.96°$

    $\beta=\arctan(g'(x))$
    $\beta=\arctan(1)=45°$

  4. Specify the angle of intersection

    $\gamma_1=|\alpha-\beta|$
    $\gamma_1=|75.96°-45°|$ $=30.96°$

    $\gamma_2=180°-|\alpha-\beta|$
    $\gamma_2=180°-30.96°$ $=149.04°$

    $\gamma_1<\gamma_2$
    => The angle of intersection $\gamma$ is $30.96°$