# 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:
- $\frac{a}{\mathrm{sin}(A)}=\frac{b}{\mathrm{sin}(B)}=\frac{c}{\mathrm{sin}(C)}=2R$

- 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:

**3. Incircle and Trigonometry:**

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

- The incircle, with inradius ($r$), introduces half-angle formulas in trigonometry. For example:

**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:
- ${c}^{2}={a}^{2}+{b}^{2}-2ab\mathrm{cos}(C)$
- This rule is derived from the Law of Cosines and is related to the circumradius.

**6.** **Inradius and Trigonometry:**

- The inradius (radius of the incircle) is also connected to trigonometry through the area of the triangle.
- The area ($A$) of a triangle with inradius ($r$) and semiperimeter ($s$) is given by:
- $A=rs$