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Law of Sines and Cosines

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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 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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