# What is the equation of the tangent line of #r=-2sin(6theta-(4pi)/3) # at #theta=(2pi)/3#?

The reference Tangents with Polar Coordinates gives us the equation:

Compute

Substitute right sides of the equation for r and

The slope, m, of the tangent line is the above equation evaluated at

The x coordinate,

The y coordinate,

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

We obtain the following equation:

Here is a graph of the function, the point of tangency, and the tangent line:

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To find the equation of the tangent line of ( r = -2\sin(6\theta - \frac{4\pi}{3}) ) at ( \theta = \frac{2\pi}{3} ), follow these steps:

- Calculate the derivative of ( r ) with respect to ( \theta ), denoted as ( \frac{dr}{d\theta} ).
- Evaluate ( \frac{dr}{d\theta} ) at ( \theta = \frac{2\pi}{3} ) to find the slope of the tangent line.
- Use the point-slope form of a line ( y - y_1 = m(x - x_1) ), where ( (x_1, y_1) ) is the point of tangency and ( m ) is the slope.
- Substitute the values of ( x_1 ), ( y_1 ), and ( m ) to find the equation of the tangent line.

By performing these steps, you'll obtain the equation of the tangent line at the given point.

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