Question

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

Answer 1

Please help

Answer 2

you are looking for the slope in x-y coordinates

you need this:

\(\huge \frac{dy}{dx} = \frac{\frac{dr}{d \theta} sin \ \theta + r \ cos \ \theta} {\frac{dr}{d \theta} cos \theta - r sin \ \theta}\)

\(\large \frac{dr}{d \theta} = \frac{d}{d \theta} (sin6\theta) = 6 cos 6 \theta\)

\(\large \theta = \frac{\pi}{12} \implies \frac{dr}{d \theta} = 6 cos (6 *(\frac{\pi}{12})) =6 cos (\pi/2 ) = 0\)

\(\large r(\frac{\pi}{12}) = sin(6 \theta) = sin(6*\frac{\pi}{12}) = sin( \pi / 2) = 1\)

\(\large \frac{dy}{dx} = -cot (\frac{\pi}{12}) \)

Answer 3

So what is the slope?

Answer 4

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

Answer 5

in my example r = 4

Answer 6

This video explains the method nicely

https://www.youtube.com/watch?v=GkhOx4hUssA

Answer 7

What would it be for me

Answer 8

-3.73

i make it

steep !

Answer 9

I will give you an example one second

Answer 10

Thank u

Answer 11

again \( \theta\) can be blagged because \(\theta = \pi / 12\) sits on the extreme of a petal and on the circle [of radius 1]

Answer 12

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

Answer 13

Here is the explicit answer just hit approximate for result and you will see the result I recommend you check it out

Answer 14

\(r=sin(6\theta)~ and~ \theta=\pi/12, so~ r=sin(6\pi/12)=sin(\pi/2)=1\)