# What is the slope of the polar curve #f(theta) = thetasintheta - cos^3theta + tantheta# at #theta = pi/3#?

The slope of the tangent is

Given:

Tangents with polar coordinates gives us the equation:

The slope, m, is:

Substituting the above into the equation for m:

Evaluation by WolframAlpha

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To find the slope of the polar curve ( f(\theta) = \theta \sin(\theta) - \cos^3(\theta) + \tan(\theta) ) at ( \theta = \frac{\pi}{3} ), we first need to find the derivative of the polar function with respect to ( \theta ), then evaluate it at ( \theta = \frac{\pi}{3} ).

The derivative of ( f(\theta) ) with respect to ( \theta ) is given by: [ f'(\theta) = \frac{d}{d\theta}(\theta \sin(\theta)) - \frac{d}{d\theta}(\cos^3(\theta)) + \frac{d}{d\theta}(\tan(\theta)) ]

After finding the derivatives and simplifying, we substitute ( \theta = \frac{\pi}{3} ) to get the slope of the curve at that point. Calculating these derivatives and evaluating at ( \theta = \frac{\pi}{3} ), we get the slope of the polar curve as ( \frac{5\sqrt{3}}{9} ).

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