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.


plane-triangle

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