Area of Triangle in Terms of Two Sides and Angle/Proof 2

Theorem
The area of a triangle $ABC$ is given by:
 * $\displaystyle \frac 1 2 a b \sin C$

where:
 * $a, b$ are two of the sides
 * $C$ is the angle of the vertex opposite $c$.

Proof

 * TriangleAreaTwoSidesAngle-2.png

By definition of sine:
 * $h = b \sin C$

From Area of Triangle in Terms of Side and Altitude:
 * $\operatorname {Area} \left({ABC}\right) = \dfrac {a h} 2$

Substituting:
 * $\operatorname {Area} \left({ABC}\right) = \dfrac {a b \sin C} 2$