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Plotting a sine rose curve by rotation of a cosine rose curve and animation of rotating polar curve.
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Plotting a sine rose curve by rotation of a cosine rose curve and animation of rotating polar curve.
In this video, we are given the polar equation of a rose curve r=3cos(5*theta). Because the coefficient of theta is odd, the rose has the same number of petals as the coefficient. We are asked to use the graph of the original rose curve to sketch the graph of r=3sin(5*theta).
To get this done, we use the trig identity sin(theta)=cos(theta-pi/2). In other words, we obtain the sine by advancing the cosine through an angle of pi/2. We apply the identity to our cosine rose, then factor out the 5 to obtain r=3cos(5(theta-pi/10)). This is a rotation of the polar curve counterclockwise through an angle of pi/10.
Finally, we use a computer algebra system to animate the rotation of the rose curve, and we obtain our picture of r=3sin(5*theta). So we successfully plot a sine rose curve using a rotation. We dramatically fade out leaving the graph of r=3sin(5*theta) in the foreground (yay, new skill!).
Plotting a sine rose curve by rotation of a cosine rose curve and animation of rotating polar curve.
In this video, we are given the polar equation of a rose curve r=3cos(5*theta). Because the coefficient of theta is odd, the rose has the same number of petals as the coefficient. We are asked to use the graph of the original rose curve to sketch the graph of r=3sin(5*theta).
To get this done, we use the trig identity sin(theta)=cos(theta-pi/2). In other words, we obtain the sine by advancing the cosine through an angle of pi/2. We apply the identity to our cosine rose, then factor out the 5 to obtain r=3cos(5(theta-pi/10)). This is a rotation of the polar curve counterclockwise through an angle of pi/10.
Finally, we use a computer algebra system to animate the rotation of the rose curve, and we obtain our picture of r=3sin(5*theta). So we successfully plot a sine rose curve using a rotation. We dramatically fade out leaving the graph of r=3sin(5*theta) in the foreground (yay, new skill!).
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