Problem
GEO-B3-M06-P021 Formula for \(\cos A+\cos B+\cos C\)
#21
★★★★☆ Level 4 of 5
Prove that in triangle \(ABC\), \(\cos A+\cos B+\cos C=1+\frac rR\), where \(r\) and \(R\) are the inradius and circumradius.
Inspired by Prasolov trigonometric geometry method
C. Hint 1. Use \(\cos A=1-2\sin^2\frac A2\) or formulas through \(p,r,R\).
D. Hint 2. It is useful to know \(\sin\frac A2\sin\frac B2\sin\frac C2=\frac r{4R}\).
Use the triangle-angle identity \[ \cos A+\cos B+\cos C=1+4\sin\frac A2\sin\frac B2\sin\frac C2. \] It follows from sum-to-product formulas and \(A+B+C=180^\circ\).
Now apply the standard formula \(\sin\frac A2\sin\frac B2\sin\frac C2=\frac r{4R}\). Hence \(\cos A+\cos B+\cos C=1+\frac rR\).
A basic trigonometric formula for problems involving \(r\) and \(R\).