Trigonometry! #69

Geometry Level 3

If A , B A,B and C C are angles of an acute angled triangle, then the minimum value of tan ⁡ 4 A + tan ⁡ 4 B + tan ⁡ 4 C \tan^{4} A + \tan^{4} B + \tan^{4} C is?

This problem is part of the set Trigonometry .


The answer is 27.

Omkar Kulkarni by Omkar Kulkarni , 22, India

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2 solutions

Raj Rajput
Feb 4, 2016

4 Helpful 0 Interesting 0 Brilliant 0 Confused
Raushan Sharma
Feb 12, 2015

The value is minimum when A=B=C=60. So the value will be 9+9+9=27

0 Helpful 0 Interesting 0 Brilliant 0 Confused

Why will the minimum value occur at 6 0 ∘ 60^{\circ} ?

Omkar Kulkarni - 6 years, 4 months ago

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Actually, it comes from the AM-HM inequality. AM is the least when AM=HM.Then simplifying that u will get that equality for the least value of AM will hold only when A=B=C. Or else, the best solution is to use the Jensen's inequality

Raushan Sharma - 6 years, 4 months ago

Because of symmetry.

Vilim Lendvaj - 2 years, 6 months ago

Because 🔺 is equilateral

Kumar Krish - 2 years, 5 months ago

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