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}}