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Input a combination of 3 angles and sides. The rest of the triangle will be solved, with all work shown.

### Law of Sines

Used when you know two angles and the included side (ASA), two angles and the non-included side (AAS) or two sides and the non-included angle (SSA).

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

Solve by cross multiplying.

$b\sin{A} = a\sin{B}$

Solve for a side:

$b = \frac{a\sin{B}}{\sin{A}}$

Or solve for an angle:

$\sin{A} = \frac{a\sin{B}}{b}$

The angles of a triangle must add to $180^{\circ}{\rm.}$ When you know two, find the third by subtracting from 180.

$A + B + C = 180^{\circ}$

### Law of Cosines

Used when you know three sides (SSS) or two sides and the included angle (SAS).

$a^2 = b^2 + c^2 - 2bc\cos{A}$ $b^2 = a^2 + c^2 - 2ac\cos{B}$ $c^2 = a^2 + b^2 - 2ab\cos{C}$

When solving for missing angles, the equations can be rearranged:

$\cos{A} = \frac{b^2 + c^2 - a^2}{2bc}$ $\cos{B} = \frac{a^2 + c^2 - b^2}{2ac}$ $\cos{C} = \frac{a^2 + b^2 -c^2}{2ab}$

The angles of a triangle must add to $180^{\circ}{\rm.}$ When you know two, find the third by subtracting from 180.

$A + B + C = 180^{\circ}$