Question

Sketch the given curves and find their points of intersection. $$ r=3 \sqrt{3} \cos \theta, r=3 \sin \theta $$

Short answer

The curves intersect at the origin (0,0).

Step-by-step solution

Identify the Equations

We are given two polar equations: 1. \( r = 3 \sqrt{3} \cos \theta \)2. \( r = 3 \sin \theta \). We will work with these equations to sketch the curves and find their points of intersection.

Convert to Cartesian Coordinates

Convert the polar equations to Cartesian coordinates using the transformations \( x = r \cos \theta \) and \( y = r \sin \theta \). For \( r = 3 \sqrt{3} \cos \theta \):\[\begin{align*}x &= r \cos \theta = 3 \sqrt{3} \cos \theta \cdot \cos \theta = 3 \sqrt{3} \cos^2 \theta,onumber\y &= r \sin \theta = 3 \sqrt{3} \cos \theta \cdot \sin \theta = 3 \sqrt{3} \cos \theta \sin \theta.\end{align*}\]For \( r = 3 \sin \theta \):\[\begin{align*}x &= r \cos \theta = 3 \sin \theta \cdot \cos \theta = 3 \sin \theta \cos \theta,onumber\y &= r \sin \theta = 3 \sin^2 \theta.\end{align*}\]

Simplify the Equations

Equate the rectangular forms obtained for each equation to simplify:The first equation becomes a circle:\( x^2 + y^2 = (3\sqrt{3} \cos \theta)^2 = 27 \cos^2 \theta \).The second equation becomes a circle:\( x^2 + y^2 = (3 \sin \theta)^2 = 9 \sin^2 \theta \).By substituting \( x \) and \( y \) with \( r \cos \theta \) and \( r \sin \theta \):\( x^2 + y^2 = r^2 = 27 \).\( x^2 + y^2 = r^2 = 9 \).

Set Equations Equal to Find Intersection Points

Solve the equations from Step 3 to find intersection points:Since both are circle equations:\[ r^2 = 27 \] and \[ r^2 = 9 \], they intersect when \( r = 0 \), which trivially is the origin, since both can't be valid simultaneously except at the origin.

Plot the Curves

Plot the curves using the equations:- For \( r = 3 \sqrt{3} \cos \theta \): This is a horizontal line.- For \( r = 3 \sin \theta \): This is a vertical line.Since \( r = 0 \) is a trivial solution, it's clear that they intersect at the origin \((0,0)\), confirming only one common solution point.

Key concepts

Polar Coordinates

Polar coordinates offer a unique way to represent points in a plane. Unlike the Cartesian coordinate system, which uses a pair of numbers to indicate distances along horizontal and vertical axes, polar coordinates use a distance and an angle. In this system, the first number is the radius \( r \) — the distance from the origin — while the second is the angle \( \theta \) — measured in radians from the positive x-axis.

This system is particularly useful in situations where symmetries around a central point are involved. Many mathematical curves have simple polar equations, which would be complicated in Cartesian form. Polar coordinates are vital in calculus as they allow for simplifications and insights into certain types of problems, especially those involving spirals, circles, and other curves best visualized with angles and distances from a central point.

Points of Intersection

Finding points of intersection is crucial for understanding how curves interact in a plane. With polar equations, the task often involves setting two equations equal to one another to see where they coincide. The process typically involves solving coordinate pairs \((r, \theta)\) where the curves share points.

In the problem provided, the equations \( r = 3 \sqrt{3} \cos \theta \) and \( r = 3 \sin \theta \) describe two curves. When solving for intersection, we observe that the intersection occurs where both equations agree — which in this scenario is when \( r = 0 \), meaning they intersect at the origin \((0,0)\).

Understanding how to convert these polar equations to rectangular form (Cartesian coordinates) can sometimes simplify the identification of intersections, providing clearer insights into their geometric nature.

Cartesian Coordinates

Cartesian coordinates are the standard way to describe points in a plane using a horizontal axis \( x \) and a vertical axis \( y \). They allow us to represent different types of geometric shapes such as lines, circles, and ellipses using equations.

In the given exercise, we convert polar equations to Cartesian coordinates to gain a clear picture of the shape and interaction of the curves. Conversion formulas are:

• \( x = r \cos \theta \)
• \( y = r \sin \theta \)

This translation simplifies further analysis, particularly in looking for intersections or symmetries, and is an essential skill in calculus. It enables us to evaluate and integrate functions over areas that are simpler to define using polar equations but analyzed using Cartesian techniques.

Curve Sketching

Curve sketching involves creating a graphical representation of equations on a coordinate system. It's a fundamental technique in calculus, helping students understand the behavior of functions and their shapes.

For polar and Cartesian equations, sketching can reveal where curves might meet or overlap, a critical step when seeking intersections. It also displays visual aspects of the graph that might be less obvious algebraically, such as curvature, symmetry, and periodicity.

In your specific exercise, sketching the polar curves shows how \( r = 3 \sqrt{3} \cos \theta \) forms a line with a unique behavior, while \( r = 3 \sin \theta \) does the same in another orientation. Observing these sketches guides the understanding of their meeting point at the origin \((0,0)\), offering a concrete preparatory step towards more complex calculus applications. Curve sketching unifies algebraic equations with their graphical counterparts, embedding a deeper comprehension of function dynamics.