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Introduction to Points, Vectors, And Functions - At A Glance:

We need one more tool to solve for intersections of polar functions. We need to know how to solve a system of two equations.

We can find where two rectangular functions y = (x) and y = g(x) intersect by setting them equal to each other and solving the equation (x) = g(x) for x. This is the place (or places) where both equations are true at the same time.

Like the intersection of two rectangular functions, we can find most of the places two polar equations r = (θ) and r = g(θ) intersect by setting them equal to each other and solving the equation (θ) = g(θ) for θ.

It's useful to graph the functions before finding where they intersect. We want to know how many intersection points we need to find. Also, setting equations equal to each other and solving might not catch an intersection point at the origin.

This sort of thing can happen because with polar coordinates there are infinitely many ways to write any point. Pick your favorite and stop there.

When finding where two polar graphs intersect, graph the functions first. Then look at how many intersection points there are and which quadrants they're in. Then we'll know how many points we need to find and roughly where they are.

We can estimate the intersection points from the graph. The graphs of r = cos θ and r = sinθ do look like they hit each other around θ = π / 4. However, we still need to set the functions equal to each other and solve for θ.

Example 1

Find the points at which r = 1 and r = sin θ intersect.


Example 2

Find where r = cos θ and r = sin θ intersect.


Exercise 1

  • Find all points where r = 1 + cosθ and r = 1 + sin θ intersect.

Exercise 2

  • Find all points where r = 1 + cos θ and r = -cos θ intersect.

Exercise 3

  • Find all points in the first quadrant where r = 2sin(3θ) and r = 1 intersect. 
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