# What is the slope of the tangent line of #r=theta/3+sin((3theta)/8-(5pi)/3)# at #theta=(7pi)/6#?

Luckily, we can apply a version of the chain rule which states that

We will also have to use the rectangular --> polar coordinate formulas:

Now we can find our slope:

Final Answer

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To find the slope of the tangent line of ( r = \frac{\theta}{3} + \sin\left(\frac{3\theta}{8} - \frac{5\pi}{3}\right) ) at ( \theta = \frac{7\pi}{6} ), you need to find the derivative ( \frac{dr}{d\theta} ) and then evaluate it at ( \theta = \frac{7\pi}{6} ).

First, differentiate ( r ) with respect to ( \theta ):

[ \frac{dr}{d\theta} = \frac{1}{3} + \frac{3}{8}\cos\left(\frac{3\theta}{8} - \frac{5\pi}{3}\right) ]

Now, evaluate ( \frac{dr}{d\theta} ) at ( \theta = \frac{7\pi}{6} ):

[ \frac{dr}{d\theta}\Bigg|_{\theta = \frac{7\pi}{6}} = \frac{1}{3} + \frac{3}{8}\cos\left(\frac{3}{8}\left(\frac{7\pi}{6}\right) - \frac{5\pi}{3}\right) ]

[ = \frac{1}{3} + \frac{3}{8}\cos\left(\frac{7\pi}{8} - \frac{5\pi}{3}\right) ]

[ = \frac{1}{3} + \frac{3}{8}\cos\left(\frac{21\pi}{24} - \frac{40\pi}{24}\right) ]

[ = \frac{1}{3} + \frac{3}{8}\cos\left(\frac{-19\pi}{24}\right) ]

[ = \frac{1}{3} + \frac{3}{8}\cos\left(-\frac{19\pi}{24}\right) ]

[ = \frac{1}{3} + \frac{3}{8}\cos\left(\frac{5\pi}{24}\right) ]

Now, compute the cosine of ( \frac{5\pi}{24} ), then substitute it into the expression:

[ \cos\left(\frac{5\pi}{24}\right) \approx 0.9659 ]

[ \frac{dr}{d\theta}\Bigg|_{\theta = \frac{7\pi}{6}} \approx \frac{1}{3} + \frac{3}{8} \times 0.9659 ]

[ \approx \frac{1}{3} + 0.3629 ]

[ \approx 0.6959 ]

Therefore, the slope of the tangent line at ( \theta = \frac{7\pi}{6} ) is approximately ( 0.6959 ).

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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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