Question

In a plane triangle find the maximum value of \( \cos A \cdot \cos B \cdot \cos C \).

Answer 1

\(\because\) A.M. \(\geq\) G.M.

\(\therefore\) \(\frac{cos A+cos B+cos C}3\geq(cos A. cos B. cos C)^{1/3}\)

But we know that Cos A + Cos B + Cos C \(\leq\) 3/2

\(\therefore\) (cos A. Cos B. Cos C)^1/3 \(\leq\) \(\frac{3/2}3=\frac12\)

\(\therefore\) Cos A. Cos B. Cos C \(\leq\) \((\frac1{2})^3\) 

⇒ Cos A. Cos B. Cos C \(\leq\) \(\frac18\) 

Hence, maximum value of Cos A. Cos B. Cos C is 1/8