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

dy/d theta = -a sin theta * sin theta + a(1 + cos theta) cos theta

= a(cos 2 theta + cos theta) (after simplification using cos^2 - sin^2 = cos 2 theta)

= a * 2 cos(3 theta/2) cos(theta/2) (sum-to-product formula)

dx/d theta = -a sin theta (1 + 2 cos theta)

Horizontal tangents (dy/d theta = 0):

cos(3 theta/2) = 0 gives theta = pi/3, pi, 5pi/3

cos(theta/2) = 0 gives theta = pi

At theta = pi, both numerator and denominator vanish (the pole) -- handle separately.

The horizontal tangents are at theta = pi/3 and theta = 5pi/3, where r = 3a/2.

Vertical tangents (dx/d theta = 0):

sin theta = 0 gives theta = 0 (r = 2a) and theta = pi (pole).

1 + 2 cos theta = 0 gives theta = 2pi/3 and theta = 4pi/3.

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4 Worked Example 2 -- Circle

Find dy/dx for r = 2 cos theta at theta = pi/4.

Solution:

dr/d theta = -2 sin theta

dy/d theta = 2 cos 2 theta

dx/d theta = -2 sin 2 theta

dy/dx = -cot 2 theta

At theta = pi/4: dy/dx = -cot(pi/2) = 0

The tangent is horizontal at theta = pi/4, which corresponds to the top of the circle.

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5 Tangent Lines at the Pole

If r = f(theta) and f(theta_0) = 0 but f'(theta_0) is not 0, then the tangent to the curve at the pole is the line theta = theta_0.

Worked Example 3

Find the tangent lines at the pole for r = sin(2 theta).

r = 0 when sin(2 theta) = 0, giving theta = 0, pi/2, pi, 3pi/2.