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Geometry Level 3

Determine the value of ∑ n = 1 89 1 1 + cot ⁡ ( n ∘ ) \displaystyle \large \sum_{n=1}^{89} \frac 1{1+\cot (n^\circ)} .


The answer is 44.5.

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Keshav Tiwari by Keshav Tiwari , 23, India

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1 solution

f ( x ) = 1 1 + cot ⁡ x = sin ⁡ x sin ⁡ x + cos ⁡ x f\left(x\right) = \dfrac{1}{1+\cot x}=\dfrac{\sin x}{\sin x+ \cos x} Clearly, f ( x ) + f ( 90 − x ) = 1 f\left(x\right) + f\left(90-x\right) =1 Therefore,the required sum comes down to - 44 + f ( 45 ) 44+ f(45) = 44.5 44.5

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LaTeX tip: Write \sin x instead of sin x . (This gives sin ⁡ x \sin x instead of s i n x sin x .) Similarly for \cos , \cot , \log , \ln , \sinh , \inf , \sup , and many more.

Akiva Weinberger - 6 years, 1 month ago

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Thanks for the information :)

Siddharth Bhatnagar - 6 years, 1 month ago

Good work ! Upvoted ! : ) :)

Keshav Tiwari - 6 years, 1 month ago

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