Ratio of cosine sums to sine sums

Geometry Level 4

sin ⁡ 2 x 1 + sin ⁡ 2 x 2 + . . . + sin ⁡ 2 x 10 = 1 \sin^{2}x_{1}+\sin^{2}x_{2}+...+\sin^{2}x_{10}=1

Let x 1 , x 2 . . . . . x 10 x_{1},x_{2}.....x_{10} be real numbers in the interval [ 0 , π 2 ] [0,\frac{\pi}{2}] such that the equation above is satisfied.

If cos ⁡ x 1 + cos ⁡ x 2 + . . . . + cos ⁡ x 10 sin ⁡ x 1 + sin ⁡ x 2 + . . . . + sin ⁡ x 10 ≥ α \frac{\cos x_{1}+\cos x_{2}+....+\cos x_{10}}{\sin x_{1}+\sin x_{2}+....+\sin x_{10}} \ge \alpha then find maximum value of α \alpha .


The answer is 3.

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Tanishq Varshney by Tanishq Varshney , 23, India

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1 solution

Saswat Prakash
Oct 3, 2017

i think the answer isnt 3 bcz if we put x1=x2=x3=x4=x5=x6=x7=x8=x9=0 ;x10=90degree then the given expression becomes 9, so 9 must be the ans

1 Helpful 0 Interesting 0 Brilliant 0 Confused

You have to find the maximum value of alpha such that the inequality always hold true . You have found out that the expression equals to 9 at some particular value. 3 is the answer because the expression is always greater than 3.

Ankit Kumar Jain - 3 years ago

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