#
Law of Sines and Cosines

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

Law of Sines Law of Cosines

### 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.

\[
A + B + C = 180^{\circ}
\]

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### 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.

\[
A + B + C = 180^{\circ}
\]

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