# What is the slope of the tangent line of #r=thetacos(theta/4-(5pi)/3)# at #theta=(pi)/3#?

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To find the slope of the tangent line of the polar curve ( r = \theta \cos\left(\frac{\theta}{4} - \frac{5\pi}{3}\right) ) at ( \theta = \frac{\pi}{3} ), you need to find the derivative ( \frac{dr}{d\theta} ) and then evaluate it at ( \theta = \frac{\pi}{3} ).

Differentiating ( r ) with respect to ( \theta ), you get:

[ \frac{dr}{d\theta} = \cos\left(\frac{\theta}{4} - \frac{5\pi}{3}\right) - \theta\sin\left(\frac{\theta}{4} - \frac{5\pi}{3}\right)\left(\frac{1}{4}\right) ]

Now, evaluate this expression at ( \theta = \frac{\pi}{3} ) to find the slope of the tangent line at that point.

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