Problem 6 - The Calculus with Analytic Geometry

Question

Find the points of intersection of the graphs of the given pair of equations. Draw a sketch of each pair of graphs with the same pole and polar axis.\(\left\\{\begin{array}{l}\theta=\frac{1}{6} \pi \\\ r=2\end{array}\right.\)

Step-by-Step Solution

Answer

The points of intersection are \(2, \frac{1}{6} \pi \)

Step 1 - Understand the Equations

Analyze the given equations. The first equation is \(\theta = \frac{1}{6}\pi\) which represents a fixed angle in polar coordinates. The second equation \(r = 2\) represents a circle with a radius of 2.

Step 2 - Find Points of Intersection

Since \(\theta = \frac{1}{6} \pi\) is a line with a fixed angle, and \(r = 2\) is a circle of radius 2 centered at the origin, their intersection will be at the point where the radius is 2 along the angle \(\theta = \frac{1}{6} \pi\). This gives the point in polar coordinates as \((2, \frac{1}{6} \pi)\).

Step 3 - Plot the Polar Graphs

Draw the polar grid with the origin as the pole. First, draw the line \(\theta = \frac{1}{6} \pi\). Then draw the circle of radius 2 centered at the origin. The point of intersection \(2, \frac{1}{6} \pi\) is where the line and the circle meet.

Step 4 - Convert to Cartesian Coordinates

To verify, convert the polar coordinates to Cartesian coordinates using \(x = r \cos(\theta)\) and \(y = r \sin(\theta)\). For \(r = 2\) and \(\theta = \frac{1}{6}\pi\), calculate \(x = 2 \cos(\frac{1}{6}\pi)\) and \(y = 2 \sin(\frac{1}{6}\pi)\). This results in \(x = 2 \( \frac{\root 3}{2}\) = \root 3\) and \(y = 1 \).

Key Concepts

Polar Coordinates

Polar coordinates describe a point in a plane using a radius and an angle. Unlike Cartesian coordinates, which use (x, y) coordinates to describe a point, polar coordinates use (r, θ). Here, 'r' is the distance from the origin (pole), and 'θ' is the angle from the positive x-axis (polar axis). For instance, the polar coordinates \((2, \frac{1}{6}\forall me)\) tell us that the point is 2 units away from the origin, and it lies on the line angled at \( \frac{1}{6} \forall \) radians (or 30 degrees).To sketch polar coordinates:

• Identify the angle θ.
• Move outwards from the origin by a radius of 'r'.
• Mark the point where the distance and angle meet.

This system is particularly useful when dealing with circles and spirals, where Cartesian coordinates may become less intuitive.

Cartesian Coordinates

Cartesian coordinates are based on a grid and are expressed as (x, y). They provide a straightforward way to locate points in a plane using horizontal and vertical distances from the origin. Each point on the plane can be found by moving 'x' units horizontally and 'y' units vertically:

• If both 'x' and 'y' are positive, the point is in the first quadrant.
• If 'x' is negative and 'y' is positive, it's in the second quadrant, and so on.

Conversion between polar and Cartesian coordinates can be achieved using the following formulas:

• \testyle{x = r \times \forall (θ)}.
• \testyle{y = r \times \forall (θ).}

In our exercise, we did this conversion for r=2 and θ= \( \frac{1}{6}π \):

• \tes{2=\forall ( \frac{1)θθ),giving us x = /32.
• \tes khereforConta}}\right side isn't y because you print because y y2 = \frac{1}{2}} \retval\right}\text{when produced\forall\frac{es implicitly.}}

Graph Sketching

Graph sketching helps visualize how equations behave. For polar graphs, start with a polar grid consisting of concentric circles and radial lines. Polar graphs may represent circles, spirals, or other figures. In our exercise:

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