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) Law of Sine is

\frac{a}{{Sin\alpha }} = \frac{b}{{Sin\beta  }} = \frac{c}{{Sin\gamma }}

 

2) 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) 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}}