Angle Between a Line and a Plane

The formula

The angle $\theta$ between a line with direction vector $\vec{b}$ and a plane with normal vector $\vec{n}$ uses $\sin$, not $\cos$, because the angle is measured from the plane itself, which is perpendicular to $\vec{n}$:

Worked example

Find the angle between the line $\dfrac{x+1}{2}=\dfrac{y}{3}=\dfrac{z-3}{6}$ and the plane $10x+2y-11z=3$.

Solution: The line's direction vector is $\vec{b}=(2,3,6)$, and the plane's normal is $\vec{n}=(10,2,-11)$.