If in a triangle ABC, cos A + cos B + cos C= 3/2 Prove that triangle ABC is - Maths - Solution of Triangles

Question

If in a triangle ABC, cos A + cos B + cos C= 3/2 Prove that
triangle ABC is an equilateral triangle

Rahul Raj · Accepted Answer

dear student

cosA+cosB+cosC=3/22(2cosA+B2cosA-B2)+2cosC=32(2cosπ2-C2cosA-B2)+2(1-2sin2(C/2))=34sin2(C/2)-4sin(C/2)cosA-B2+1=0This is a quadratic equation in sinc/2, and it has real roots Therefore , Descriminant ≥0 16cos2A-B2-16≥0cos2A-B2≥1so cos2A-B2=1 as cosine is always ≤1only true when A=Bsimilarly we can show B=Cand triangle is equilateral

regards

If in a triangle ABC, cos A + cos B + cos C= 3/2 .Prove that triangle ABC is an equilateral triangle .