# Results and Formulas of Plane Triangles

Consider the triangle $\Delta ABC$, having the angles $\alpha ,\beta ,\gamma$ and sides $a,b,c$ as shown in the figure.

1) The law of sine is $\frac{a}{{Sin\alpha }} = \frac{b}{{Sin\beta }} = \frac{c}{{Sin\gamma }}$

2) The laws of cosine are

${a^2} = {b^2} + {c^2} – 2bcCos\alpha$ or $Cos\alpha = \frac{{{b^2} + {c^2} – {a^2}}}{{2bc}}$

${b^2} = {a^2} + {c^2} – 2acCos\beta$ or $Cos\beta = \frac{{{a^2} + {c^2} – {b^2}}}{{2ac}}$

${c^2} = {a^2} + {b^2} – 2abCos\gamma$ or $Cos\gamma = \frac{{{a^2} + {b^2} – {c^2}}}{{2ab}}$

3) The laws of tangent are

$\frac{{Tan\left( {\frac{{A – B}}{2}} \right)}}{{Tan\left( {\frac{{A + B}}{2}} \right)}} = \frac{{a – b}}{{a + b}}$

$\frac{{Tan\left( {\frac{{B – C}}{2}} \right)}}{{Tan\left( {\frac{{B + C}}{2}} \right)}} = \frac{{b – c}}{{b + c}}$

$\frac{{Tan\left( {\frac{{A – C}}{2}} \right)}}{{Tan\left( {\frac{{A + C}}{2}} \right)}} = \frac{{a – c}}{{a + c}}$

HALF ANGLE FORMULAS

Consider that $a,b,c$ and $A,B,C$ are sides and angles of $\Delta ABC$, as shown above. Also if (i) $s = \frac{{a + b + c}}{2}$ (ii) $r = \sqrt {\frac{{(s – a)(s – b)(s – c)}}{s}}$

4) $Sin\frac{\alpha }{2} = \sqrt {\frac{{(s – b)(s – c)}}{{bc}}}$

5) $Sin\frac{\beta }{2} = \sqrt {\frac{{(s – a)(s – c)}}{{ac}}}$

6) $Sin\frac{\gamma }{2} = \sqrt {\frac{{(s – a)(s – b)}}{{ab}}}$

7) $Cos\frac{\alpha }{2} = \sqrt {\frac{{s{\text{ }}(s – a)}}{{bc}}}$

8) $Cos\frac{\beta }{2} = \sqrt {\frac{{s{\text{ }}(s – b)}}{{ac}}}$

9) $Cos\frac{\gamma }{2} = \sqrt {\frac{{s{\text{ }}(s – c)}}{{ab}}}$

10) $Tan\frac{\alpha }{2} = \sqrt {\frac{{(s – b)(s – c)}}{{s{\text{ }}(s – a)}}} = \frac{r}{{s – a}}$

11) $Tan\frac{\beta }{2} = \sqrt {\frac{{(s – a)(s – c)}}{{s{\text{ }}(s – b)}}} = \frac{r}{{s – b}}$

12) $Tan\frac{\gamma }{2} = \sqrt {\frac{{(s – a)(s – b)}}{{s{\text{ }}(s – c)}}} = \frac{r}{{s – c}}$