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anonymous

  • one year ago

Slope of the tangent line to the polar curve r=sin(6theta). Theta =pi/12. Find slope

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  1. anonymous
    • one year ago
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    Please help

  2. IrishBoy123
    • one year ago
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    you are looking for the slope in x-y coordinates you need this: \(\huge \frac{dy}{dx} = \frac{\frac{dr}{d \theta} sin \ \theta + r \ cos \ \theta} {\frac{dr}{d \theta} cos \theta - r sin \ \theta}\) \(\large \frac{dr}{d \theta} = \frac{d}{d \theta} (sin6\theta) = 6 cos 6 \theta\) \(\large \theta = \frac{\pi}{12} \implies \frac{dr}{d \theta} = 6 cos (6 *(\frac{\pi}{12})) =6 cos (\pi/2 ) = 0\) \(\large r(\frac{\pi}{12}) = sin(6 \theta) = sin(6*\frac{\pi}{12}) = sin( \pi / 2) = 1\) \(\large \frac{dy}{dx} = -cot (\frac{\pi}{12}) \)

  3. anonymous
    • one year ago
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    So what is the slope?

  4. Australopithecus
    • one year ago
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    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

  5. Australopithecus
    • one year ago
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    in my example r = 4

  6. Australopithecus
    • one year ago
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    This video explains the method nicely https://www.youtube.com/watch?v=GkhOx4hUssA

  7. anonymous
    • one year ago
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    What would it be for me

  8. IrishBoy123
    • one year ago
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    -3.73 i make it steep !

  9. Australopithecus
    • one year ago
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    I will give you an example one second

  10. anonymous
    • one year ago
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    Thank u

  11. IrishBoy123
    • one year ago
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    again \( \theta\) can be blagged because \(\theta = \pi / 12\) sits on the extreme of a petal and on the circle [of radius 1]

  12. Australopithecus
    • one year ago
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    Here is the explicit answer just hit approximate for result and you will see the result I recommend you check it out

  13. mathmate
    • one year ago
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    \(r=sin(6\theta)~ and~ \theta=\pi/12, so~ r=sin(6\pi/12)=sin(\pi/2)=1\)

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