# What is the equation of the tangent line of #r=theta-sin(-theta+(pi)/3) # at #theta=(2pi)/3#?

We need the Cartesian coordinates of the point of tangency:

The slope, m, of the tangent line is:

Use the point-slope form of the equation of a line:

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To find the equation of the tangent line of (r = \theta - \sin\left(-\theta + \frac{\pi}{3}\right)) at (\theta = \frac{2\pi}{3}), we need to find the slope of the tangent line at that point and the point where it touches the curve.

First, differentiate (r) with respect to (\theta) to find the slope of the tangent line: [\frac{dr}{d\theta} = 1 - (-\cos(-\theta + \frac{\pi}{3}))] [\frac{dr}{d\theta} = 1 + \cos(\theta - \frac{\pi}{3})]

Evaluate (\frac{dr}{d\theta}) at (\theta = \frac{2\pi}{3}): [\frac{dr}{d\theta} \Bigg|_{\theta = \frac{2\pi}{3}} = 1 + \cos\left(\frac{2\pi}{3} - \frac{\pi}{3}\right) = 1 + \cos(\frac{\pi}{3}) = 1 + \frac{1}{2} = \frac{3}{2}]

This is the slope of the tangent line at (\theta = \frac{2\pi}{3}).

Next, plug (\theta = \frac{2\pi}{3}) into the equation (r = \theta - \sin\left(-\theta + \frac{\pi}{3}\right)) to find the corresponding (r) value: [r \Bigg|_{\theta = \frac{2\pi}{3}} = \frac{2\pi}{3} - \sin\left(-\frac{2\pi}{3} + \frac{\pi}{3}\right) = \frac{2\pi}{3} - \sin(-\pi) = \frac{2\pi}{3}]

So, at (\theta = \frac{2\pi}{3}), (r = \frac{2\pi}{3}).

Now, we have the slope ((m = \frac{3}{2})) and a point ((\frac{2\pi}{3}, \frac{2\pi}{3})) on the tangent line. Using the point-slope form of the equation of a line, we can write the equation of the tangent line: [y - y_1 = m(x - x_1)] [y - \frac{2\pi}{3} = \frac{3}{2}\left(x - \frac{2\pi}{3}\right)]

This equation can be simplified further if desired.

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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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