Solving General Triangles

The law of sines

$\displaystyle {\frac{{a}}{{\sin \alpha }}}$ = $\displaystyle {\frac{{b}}{{\sin \beta }}}$ = $\displaystyle {\frac{{c}}{{\sin \gamma }}}$

enables you to solve a triangle if you are given one side and two angles, or if you are given two sides and an angle opposite one of these sides.dtbpF2.3298in1.5489in0pttriangle.wmf


\begin{example} To solve a triangle given one side and two angles, \par \begin{... ...qrt{3}% \sin \frac{2}{9}\pi \end{displaymath} \end{enumerate} \end{example}

You can apply Solve + Numeric to get numerical solutions, or you can evaluate the preceding solutions numerically.

Using both the law of sines and the law of cosines,

a2 + b2 -2ab cosγ = c2

you can solve a triangle given two sides and the included angle, or given three sides.


\begin{example} To solve a triangle given two sides and the included angle, \pa... ... to get $\alpha =.58859$\ and $\beta =1.0104$. \end{enumerate} \end{example}

A triangle with three sides given is solved similarly: interchange the actions on γ and c in the steps just described.


\begin{example} To solve a triangle given three sides, \par \begin{enumerate} ... ...dfrac{c}{\sin \gamma }$\ to get $\beta =.49948$. \end{enumerate} \end{example}