Trigonometric Relations in a Triangle with Circumcircle and Incircle

1. Introduction:

• Trigonometry plays a crucial role in understanding the relationships between angles and sides in a triangle. In this context, the circumcircle and incircle of a triangle bring forth interesting trigonometric relations.

2. Circumcircle and Trigonometry:

• Law of Sines:
• For any triangle with sides $a$, $b$, and $c$ and angles $A$, $B$, and $C$, the Law of Sines relates the angles and sides to the circumradius ($R$) of the circumcircle:
• $\frac{a}{\mathrm{sin}\left(A\right)}=\frac{b}{\mathrm{sin}\left(B\right)}=\frac{c}{\mathrm{sin}\left(C\right)}=2R$

3. Incircle and Trigonometry:

• Half-Angle Formulas:
• The incircle, with inradius ($r$), introduces half-angle formulas in trigonometry. For example:
• $\mathrm{tan}\left(\frac{A}{2}\right)=\frac{r}{s-a}$

4. Euler's Formula and Trigonometry:

• Euler's Formula, connecting circumradius ($R$), inradius ($r$), and the distance ($d$) between circumcenter and incenter, involves trigonometric functions:
• $R=\sqrt{{R}^{2}-2rR}+\sqrt{{R}^{2}-2rR+{d}^{2}}$

5.  Cosine Rule and Circumcircle:

• The Cosine Rule, which relates the sides and angles of a triangle, can also be connected to the circumcircle. For any triangle with sides $a$, $b$, and $c$ and corresponding angles $A$, $B$, and $C$, the Cosine Rule is given by:
• ${c}^{2}={a}^{2}+{b}^{2}-2ab\mathrm{cos}\left(C\right)$
• This rule is derived from the Law of Cosines and is related to the circumradius.

• The area ($A$) of a triangle with inradius ($r$) and semiperimeter ($s$) is given by:
• $A=rs$