Tangents to Polar Curves

Finding the gradient of a polar curve and determining the equations of tangents -- particularly horizontal and vertical tangents -- is an important skill in AQA Further Mathematics.

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1 Converting to Cartesian Derivatives

Given r = f(theta), we can express x and y in terms of theta:

x = r cos theta = f(theta) cos theta

y = r sin theta = f(theta) sin theta

Then:

dy/dx = (dy/d theta) / (dx/d theta)

where:

dy/d theta = (dr/d theta) sin theta + r cos theta

dx/d theta = (dr/d theta) cos theta - r sin theta

These formulas are obtained by applying the product rule.

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2 Horizontal and Vertical Tangents

Horizontal tangent: dy/d theta = 0 and dx/d theta is not 0.

Vertical tangent: dx/d theta = 0 and dy/d theta is not 0.

If both are zero simultaneously, you have a more complex situation (often at the pole) that requires further analysis.

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3 Worked Example 1 -- Cardioid

Find the points where r = a(1 + cos theta) has horizontal and vertical tangents.

Solution:

dr/d theta = -a sin theta