Trigonometry Cycles

Geometry Level 4

If A , B , C A,B,C are angles of a triangle on a plane surface, then M , m M,m be the maximum and minimum value of

cos ⁡ A cos ⁡ B cos ⁡ C − ∑ c y c ( cos ⁡ A sin ⁡ B sin ⁡ C ) \cos A \cos B \cos C - \displaystyle \sum_{cyc} (\cos A \sin B \sin C) .

Find M − m M-m


The answer is 0.

Nanayaranaraknas Vahdam by Nanayaranaraknas Vahdam , 22, India

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1 solution

Ronak Agarwal
Jul 6, 2014

J u s t n o t e t h a t c o s ( A + B + C ) = c o s ( A ) c o s ( B ) c o s ( C ) − ∑ c y c c o s ( A ) s i n ( B ) s i n ( C ) N o w s i n c e A + B + C = 180 ⇒ c o s ( A + B + C ) = − 1 H e n c e M = m = − 1 ⇒ M − m = 0 Just\quad note\quad that\\ cos(A+B+C)=cos(A)cos(B)cos(C)-\sum _{ cyc }^{ }{ cos(A)sin(B)sin(C) } \\ Now\quad since\quad A+B+C=180\quad \Rightarrow cos(A+B+C)=-1\\ Hence\quad M=m=-1\quad \Rightarrow M-m=0\\ \\ \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad

1 Helpful 0 Interesting 0 Brilliant 0 Confused

Just a formula based question

Kïñshük Sïñgh - 6 years, 10 months ago

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