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Geometry Level 2

Find the sum of all integers n n satisfying the following inequality 1 4 < sin ⁡ π n < 1 3 \frac{1}{4}<\sin\frac{\pi}{n}<\frac{1}{3}


The answer is 33.

Danish Ahmed by Danish Ahmed , 24, India

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1 solution

Tom Engelsman
May 3, 2021

This inequality boils down to:

π arcsin ⁡ ( 1 / 3 ) < n < π arcsin ⁡ ( 1 / 4 ) ⇒ 9.244 < n < 12.43 ⇒ n = 10 , 11 , 12. \Large \frac{\pi}{\arcsin(1/3)} < n < \frac{\pi}{\arcsin(1/4)} \Rightarrow 9.244 < n < 12.43 \Rightarrow n = 10, 11, 12.

Hence, 10 + 11 + 12 = 33 . 10+11+12 = \boxed{33}.

0 Helpful 0 Interesting 1 Brilliant 0 Confused

sin ⁡ ( x ) = n ⇒ x ∈ { arcsin ⁡ ( n ) , π − arcsin ⁡ ( n ) , − π − arcsin ⁡ ( n ) . . . } ⇒ x = arcsin ⁡ ( n ) \sin(x)=n\Rightarrow x\in \{\arcsin(n),\pi-\arcsin(n),-\pi-\arcsin(n)...\}\cancel{\Rightarrow}x=\arcsin(n)

Zakir Husain - 1 month, 1 week ago

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