# What is the equation of the tangent line of #r=cos(2theta-pi/4)/sintheta - sin(theta-pi/8)# at #theta=(-3pi)/8#?

The equation is:

Thus far we have

Evaluating the sines and cosines:

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

The equation is:

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To find the equation of the tangent line at a specific point on a polar curve, you first need to find the slope of the tangent line at that point. This is done by taking the derivative of the polar equation with respect to θ and evaluating it at the given θ value. Then, you use the point-slope form of a line to write the equation of the tangent line.

Given the polar equation ( r = \frac{\cos(2\theta - \frac{\pi}{4})}{\sin(\theta)} - \sin(\theta - \frac{\pi}{8}) ) and the point ( \theta = -\frac{3\pi}{8} ), we first find the slope of the tangent line by taking the derivative of the polar equation with respect to ( \theta ) and evaluating it at ( \theta = -\frac{3\pi}{8} ).

The derivative of the given polar equation with respect to ( \theta ) is:

[ \frac{dr}{d\theta} = \frac{-2\sin(2\theta - \frac{\pi}{4})\sin(\theta) - \cos(\theta)\cos(2\theta - \frac{\pi}{4}) - \cos(\theta - \frac{\pi}{8})}{\sin^2(\theta)} ]

Now, plug in ( \theta = -\frac{3\pi}{8} ) to find the slope of the tangent line:

[ \frac{dr}{d\theta}\bigg|_{\theta = -\frac{3\pi}{8}} ]

After evaluating, you'll have a numerical value for the slope. Let's denote this as ( m ).

Now, using the point-slope form of a line:

[ y - y_1 = m(x - x_1) ]

where ( (x_1, y_1) ) is the point on the polar curve corresponding to ( \theta = -\frac{3\pi}{8} ), which you can find by substituting ( \theta = -\frac{3\pi}{8} ) into the polar equation to get the corresponding ( r ) value.

After substituting the slope ( m ) and the point ( (x_1, y_1) ), you'll have the equation of the tangent line.

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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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