Trig Will Be Useful Here

Calculus Level 2

A point ( x , y ) \left( x,y \right) moves counterclockwise along the unit circle at constant angular speed ω \omega . Describe the motion of the point ( − 2 x y , y 2 − x 2 ) \left( -2xy, { y }^{ 2 }-{ x }^{ 2 } \right) .

clockwise along the unit circle at angular speed ω \omega clockwise along the unit circle at angular speed 2 ω 2\omega counterclockwise along the unit circle at angular speed ω \omega counterclockwise along the unit circle at angular speed 2 ω 2\omega

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Soumo Mukherjee by Soumo Mukherjee , 25, India

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1 solution

Aalap Shah
Apr 7, 2015

Since ((x,y)) is on the unit circle, we put: x = cos ⁡ ( t ) x=\cos(t) y = sin ⁡ ( t ) y=\sin(t) As t t increases, the point rotates counterclockwise. Now, using the following formulae: 2 cos ⁡ ( t ) sin ⁡ ( t ) = sin ⁡ ( 2 t ) 2\cos(t)\sin(t)=\sin(2t) cos ⁡ 2 ( t ) − sin ⁡ 2 ( t ) = cos ⁡ ( 2 t ) \cos^2(t)-\sin^2(t)=\cos(2t) We get that the other point is, in fact: ( − sin ⁡ ( 2 t ) , − cos ⁡ ( 2 t ) ) (-\sin(2t), -\cos(2t)) That is: ( cos ⁡ ( 3 π / 2 − 2 t ) , sin ⁡ ( 3 π / 2 − 2 t ) ) (\cos(3\pi /2-2t), \sin(3\pi /2-2t)) Which rotates clockwise, twice as fast as the original, since the argument is − 2 t , -2t, as t t increases.

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