Find min of

Geometry Level 2

In triangle A B C , ABC, find the maximum possible value of cos A + cos B + cos C . \cos A + \cos B + \cos C.


The answer is 1.5.

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2 solutions

Rohan Shinde
Dec 1, 2018

For maximisation all angles must be acute to give positive cosines to add up. Hence by Jensen's inequality we get the maximum value as 3 2 \frac 32 because the graph for cos \cos is concave down for acute angled.

Jeremy Galvagni
Oct 15, 2018

I accidentally got this correct. 1.5 is the maximum possible value, which occurs when the triangle is equilateral. The minimum approaches 1 as the points approach collinear.

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