29–32. Intersection points Use algebraic methods to find as many intersection points of the following curves as possible. Use graphical methods to identify the remaining intersection points.


r = 2 cos θ and r = 1 + cos θ

Verified step by step guidance

1. Start by setting the two polar equations equal to each other to find the intersection points algebraically: \[2 \cos \theta = 1 + \cos \theta\].
2. Rearrange the equation to isolate terms: \[2 \cos \theta - \cos \theta = 1 \implies \cos \theta = 1\].
3. Solve for \[\theta\] where \[\cos \theta = 1\]. Recall that \[\cos \theta = 1\] at \[\theta = 0\] (and at multiples of \[2\pi\], but consider the principal values in the interval \[[0, 2\pi)\]).
4. Substitute \[\theta = 0\] back into either original equation to find the corresponding \[r\] value for the intersection point.
5. To find any additional intersection points that may not be captured by the algebraic method, use a graphical approach by plotting both curves \[r = 2 \cos \theta\] and \[r = 1 + \cos \theta\] over the interval \[\theta \in [0, 2\pi)\] and visually identify where they intersect.

Key Concepts

Polar Coordinates and Equations

Polar coordinates represent points using a radius and an angle (r, θ) instead of Cartesian (x, y). Understanding how to interpret and manipulate polar equations like r = 2 cos θ is essential for finding points where two curves intersect.

Solving Systems of Equations

Finding intersection points involves solving the system formed by the two given equations. In this case, equate r = 2 cos θ and r = 1 + cos θ, then solve algebraically for θ and r to find common solutions representing intersection points.

Graphical Analysis of Polar Curves

Graphing polar curves helps visualize intersections that may be difficult to find algebraically, especially when multiple solutions or complex angles are involved. Plotting r = 2 cos θ and r = 1 + cos θ reveals additional intersection points and confirms algebraic results.