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

1. anonymous

2. IrishBoy123

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

3. anonymous

So what is the slope?

4. 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

5. Australopithecus

in my example r = 4

6. Australopithecus

This video explains the method nicely https://www.youtube.com/watch?v=GkhOx4hUssA

7. anonymous

What would it be for me

8. IrishBoy123

-3.73 i make it steep !

9. Australopithecus

I will give you an example one second

10. anonymous

Thank u

11. IrishBoy123

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
13. Australopithecus

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

14. mathmate

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