Tangent lines of polar curves

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ijd5000

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Sep 3, 2013
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Find the points on the given curve where the tangent line is horizontal or vertical. (Assume 0 ≤ θ ≤ 2π.
 Enter your answers as a comma-separated list of ordered pairs.)r2 = sin 2θ

converted the equation into x and y then took their derivatives, set them equal to zero and solved. The problem is i came up with the same values of θ for both the horizontal and vertical tangents. I think the problem might have happened when i took the square root of both sides of the original equation?
 
Romsek said:

\(\displaystyle r^2=\sin (2 \theta )\)

\(\displaystyle 2 r dr=2 d\theta \cos (2 \theta )\)

\(\displaystyle \frac{dr}{d\theta }=\frac{\cos (2 \theta )}{r}\)

\(\displaystyle \frac{dr}{d\theta }=\frac{\cos (2 \theta )}{\sqrt{\sin (2 \theta )}}\)


From this you can read off the horizontal tangent lines, i.e. when \(\displaystyle \cos (2 \theta )\) goes to 0
and the vertical tangent lines, i.e. when \(\displaystyle \sqrt{\sin (2 \theta )}\) goes to 0.

dy/dx in general is not equal to dr/d\(\displaystyle \theta\) but they mean the same thing when they are 0 or infinity.
Click to expand...


Horizontal tangents at pi/4 and 3pi/4 and vertical at 0, and pi/2. is still wrong. ?
 
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