Home > Calculus > Determining the Slope and Tangent Lines for a Polar Curve

What is the equation of the tangent line of #r=4cos(-theta+(5pi)/6) +sin(theta)# at #theta=(7pi)/12#?

Answer 1

Isabella Ackerman

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Answer 2

Christopher Agresta

To find the equation of the tangent line to the polar curve (r = 4\cos\left(-\theta + \frac{5\pi}{6}\right) + \sin(\theta)) at (\theta = \frac{7\pi}{12}), follow these steps:

Find the derivative of the polar curve with respect to (\theta), (dr/d\theta).
Evaluate (dr/d\theta) at (\theta = \frac{7\pi}{12}) to find the slope of the tangent line.
Use the point-slope form of the equation of a line: (y - y_1 = m(x - x_1)), where ((x_1, y_1)) is the point on the curve corresponding to (\theta = \frac{7\pi}{12}), and (m) is the slope found in step 2.
Convert the polar coordinates ((r, \theta)) of the point on the curve to Cartesian coordinates ((x, y)).
Substitute the values of (x_1), (y_1), and (m) into the point-slope form equation to find the equation of the tangent line.

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Answer from HIX Tutor

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.