Sine law and cosine law

Parametric equations

In any triangle the sides $a, b, c$ are opposite the angles $A, B, C$ . The sine law says

\frac{a}{sin A} = \frac{b}{sin B} = \frac{c}{sin C} = 2 R

where $R$ is the circumradius. The cosine law says

c^{2} = a^{2} + b^{2} - 2 ab cos C

When $C = 9 0^{\circ}$ , this reduces to the Pythagorean theorem $c^{2} = a^{2} + b^{2}$ .

Sine law: show the circumcircle and $2 R$ , and check that $a / sin A$ , $b / sin B$ , and $c / sin C$ are equal
Cosine law: highlight the included angle $C$ and the three sides, and compare $a^{2} + b^{2} - 2 ab cos C$ with $c^{2}$
Guides: on: show or hide the altitude, the foot of the perpendicular, and the circumcircle, and compare the figure with and without those guides
Drag a vertex to change the triangle. The three side lengths and the three angles update at once. Make $C$ obtuse and watch $cos C < 0$ .

The sine law ties each side to the sine of the opposite angle by one common ratio. The circumradius $R$ is the geometric quantity in that ratio, and the common value is $2 R$ .
The cosine law is the Pythagorean theorem plus a correction for the included angle. At a right angle, $cos C = 0$ , and the formula returns to $c^{2} = a^{2} + b^{2}$ .
Given two angles and one side, three sides, or two sides and the included angle, the sine law or the cosine law gives the remaining unknowns.