anonymous
  • anonymous
Slope of the tangent line to the polar curve r=sin(6theta). Theta =pi/12. Find slope
Mathematics
chestercat
  • chestercat
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anonymous
  • anonymous
Please help
IrishBoy123
  • IrishBoy123
you are looking for the slope in x-y coordinates you need this: \(\huge \frac{dy}{dx} = \frac{\frac{dr}{d \theta} sin \ \theta + r \ cos \ \theta} {\frac{dr}{d \theta} cos \theta - r sin \ \theta}\) \(\large \frac{dr}{d \theta} = \frac{d}{d \theta} (sin6\theta) = 6 cos 6 \theta\) \(\large \theta = \frac{\pi}{12} \implies \frac{dr}{d \theta} = 6 cos (6 *(\frac{\pi}{12})) =6 cos (\pi/2 ) = 0\) \(\large r(\frac{\pi}{12}) = sin(6 \theta) = sin(6*\frac{\pi}{12}) = sin( \pi / 2) = 1\) \(\large \frac{dy}{dx} = -cot (\frac{\pi}{12}) \)
anonymous
  • anonymous
So what is the slope?

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Australopithecus
  • Australopithecus
You need the formula \[\frac{dy}{dx} = \frac{\frac{dr}{d \theta} \sin(\theta) + r*\cos(\theta)}{\frac{dr}{d \theta}\cos(\theta) - r*\sin(\theta)}\] so take the derivative of sin(6theta) and plug into the formula for example if I had 4 The derivative would be 0 so I would plug that into my formula: (0*sin(theta) + 4cos(theta) )/(0*cos(theta) -4sin(theta)) Simplify then plug Theta =pi/12 into your formula and solve and you should get your derivative
Australopithecus
  • Australopithecus
in my example r = 4
Australopithecus
  • Australopithecus
This video explains the method nicely https://www.youtube.com/watch?v=GkhOx4hUssA
anonymous
  • anonymous
What would it be for me
IrishBoy123
  • IrishBoy123
-3.73 i make it steep !
Australopithecus
  • Australopithecus
I will give you an example one second
anonymous
  • anonymous
Thank u
IrishBoy123
  • IrishBoy123
again \( \theta\) can be blagged because \(\theta = \pi / 12\) sits on the extreme of a petal and on the circle [of radius 1]
Australopithecus
  • Australopithecus
http://www.wolframalpha.com/input/?i=%286cos%286x%29*sin%28x%29+%2B+sin%286x%29cos%28x%29%29%2F%286cos%286x%29cos%28x%29+-+sin%286x%29*sin%28x%29%29%2C+x+%3D+pi%2F12
Australopithecus
  • Australopithecus
Here is the explicit answer just hit approximate for result and you will see the result I recommend you check it out
mathmate
  • mathmate
\(r=sin(6\theta)~ and~ \theta=\pi/12, so~ r=sin(6\pi/12)=sin(\pi/2)=1\)

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