WMC 2018 Senior Problem 4

Level 1

Which is the smallest (in radians)?

( t a n ( 0.1 ) ) 2 { (tan(0.1)) }^{ 2 } ( s i n ( 0.1 ) ) 2 { (sin(0.1)) }^{ 2 } s i n ( s i n ( s i n ( 0.1 ) ) ) sin(sin(sin(0.1))) s i n ( s i n ( 0.1 ) ) sin(sin(0.1)) ( s i n ( 0.1 ) ) 3 { (sin(0.1)) }^{ 3 }

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1 solution

Julian Yu
Oct 4, 2018

Since 0.1 0.1 is small, we can use the truly amazing approximation:

sin x = tan x = x \large{\sin{x}=\tan{x}=x} .

Therefore, sin ( sin ( 0.1 ) ) sin ( sin ( sin ( 0.1 ) ) ) 0.1 \sin{(\sin{(0.1)})}\approx \sin{(\sin{(\sin{(0.1)})})}\approx 0.1 ,

( sin ( 0.1 ) ) 2 ( tan ( 0.1 ) ) 2 0.01 (\sin{(0.1)})^2\approx(\tan{(0.1)})^2\approx 0.01 , and

( sin ( 0.1 ) ) 3 0.001 (\sin{(0.1)})^3\approx 0.001 .

Therefore, ( sin ( 0.1 ) ) 3 \boxed{(\sin{(0.1)})^3} is the smallest.

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