How do you find the slope of the polar curve #r=3+8sin(theta)# at #theta=pi/6# ?

Answer 1

#r(theta)=3+8sin theta#

by converting into parametric equations,

#Rightarrow{(x(theta)=r(theta)cos theta=(3+8sin theta)cos theta),(y(theta)=r(theta)sin theta=(3+8sin theta)sin theta):}#

By differentiating with respect to #theta#,

#x'(theta)=8cos theta cdotcos theta+(3+8sin theta)cdot(-sin theta)#

#=8(cos^2theta-sin^2theta)-3sin theta#

#=8cos2theta-3sin theta#

by evaluating at #theta=pi/6#,

#Rightarrow x'(pi/6)=8cos(pi/3)-3sin(pi/6)=4-3/2=5/2#

By differentiating with respect to #theta#,

#y'(theta)=8cos theta cdot sin theta+(3+8sin theta)cdotcos theta#

#=16sin theta cos theta+3cos theta#

#=8sin2theta+3cos theta#

by evaluating at #theta=pi/6#,

#Rightarrow y'(pi/6)=8sin(pi/3)+3cos(pi/6)=4sqrt{3}+{3sqrt{3}}/2={11sqrt{3}}/2#

So, the slope #m# can be found by

#m={dy}/{dx}|_{theta=pi/6}={{dy}/{d theta}|_{theta = pi /6}}/{{dx]/{d theta}|_{theta = pi /6}}={y'(pi/6)}/{x'(pi/6)}={{11sqrt{3}}/{2}}/{{5}/{2}}={11sqrt{3}}/5#

The graph along with its tangent line at #theta=pi/6# looks like:


I hope that this was helpful.

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Answer 2

To find the slope of the polar curve ( r = 3 + 8\sin(\theta) ) at ( \theta = \frac{\pi}{6} ), you need to differentiate the equation with respect to ( \theta ) and then evaluate it at ( \theta = \frac{\pi}{6} ).

  1. Differentiate ( r = 3 + 8\sin(\theta) ) with respect to ( \theta ) using the chain rule.
  2. Substitute ( \theta = \frac{\pi}{6} ) into the derivative obtained in step 1 to find the slope at that specific point.
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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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