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:
      • asin(A)=bsin(B)=csin(C)=2R

3. Incircle and Trigonometry:

  • Half-Angle Formulas:
    • The incircle, with inradius (r), introduces half-angle formulas in trigonometry. For example:
      • tan(A2)=rsa

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=R22rR+R22rR+d2

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:
    • c2=a2+b22abcos(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