Problem 18 - Essential Calculus

Question

Find the points of intersection of the graphs of the equations. $$ \begin{array}{l} \theta=\frac{\pi}{4} \\ r=2 \end{array} $$

Step-by-Step Solution

Answer

The points of intersection of the graphs of the given equations are \((2\sqrt{2}, \frac{\pi}{4})\) in polar coordinates and \((2, 2)\) in Cartesian coordinates.

Step 1: Convert to Cartesian Coordinates

Let's convert our polar coordinates to Cartesian coordinates. The transformation relations are \(x = r \cos \theta\) and \(y = r \sin \theta\). For the first equation, \(\theta = \frac{\pi}{4}\), these give: \(x = r \cos \frac{\pi}{4}\), \(y = r \sin \frac{\pi}{4}\). Similarly, for the second equation, \(r = 2\), Substituting \(r = 2\) into the transformation equations gives: \(x = 2 \cos \theta\), \(y = 2 \sin \theta\).

Step 2: Find Intersection Points

Now to find the intersection points of the two graphs, set the x and y components of each equation equal to each other and solve for r: \[\begin{array}{c}1) r \cos \frac{\pi}{4} = 2 \cos \theta \2) r \sin \frac{\pi}{4} = 2 \sin \theta\end{array}\] From equation 1), we have \(r = 2 / cos \frac{\pi}{4} = \sqrt {2} * 2 = 2 \sqrt {2}\). Substituting r into equation 2) confirms this solution as the sin and cos of \(\pi/4\) are the same.

Step 3: Convert Back to Polar Coordinates

So the polar coordinates of our intersection points are \((r, \theta) = (2\sqrt{2}, \frac{\pi}{4})\). In Cartesian coordinates, these are \((x,y) = (r\cos \theta, r\sin \theta) = (2, 2)\).

Key Concepts

Intersection Points

In mathematics, intersection points occur where two or more graphs meet at common points on a coordinate plane. To find intersection points between two curves, we need to solve their equations simultaneously. In the original exercise, we are dealing with polar equations defined by \(\theta = \frac{\pi}{4}\) and \(r = 2\). The task is to find where these two polar equations intersect.The process begins by transforming the polar equations into Cartesian coordinates. This conversion facilitates the finding of intersection points, as most students are more familiar with Cartesian equations. The intersection is located where the \(x\) and \(y\) values satisfy both sets of equations simultaneously. As our step-by-step solution outlines, after converting and equalizing the Cartesian representations, solving for \(r\) and substituting back verifies our solution. The determined intersection point in polar coordinates \((2\sqrt{2}, \frac{\pi}{4})\) is equivalent to \((2, 2)\) in the Cartesian plane.

Conic Sections

Conic sections are the curves obtained by intersecting a plane with a cone. There are four primary types: circles, ellipses, parabolas, and hyperbolas, each having distinct equations in polar and Cartesian coordinates.

• A circle remains a circle in either coordinate system and is particularly simple when its center is at the origin. Its equation in polar coordinates is \(r = a\), where \(a\) is the radius.
• Ellipses have a more complex form in polar terms and are rarely reduced to a simple equation unless special conditions apply.
• Parabolas and hyperbolas show unique characteristics and would typically require more complex equations involving angles and eccentricity.

In this exercise, the equations resemble straightforward versions of these conic sections. The equation \(r = 2\) can be visualized as a circle centered at the origin with a radius \(2\), while \(\theta = \frac{\pi}{4}\) represents a line through the origin at a 45-degree angle. Such understanding of how they look graphically aids in visualizing where they might intersect.

Coordinate Transformation

Coordinate transformation involves converting equations or coordinates from one system to another, such as from polar to Cartesian coordinates or vice versa. This transformation is critical in a wide range of applications, simplifying mathematical modeling and solutions.The equations required for converting polar coordinates \((r, \theta)\) to Cartesian coordinates \((x, y)\) are:

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

By applying these formulas, the exercise's two given equations transform smoothly. For \(\theta = \frac{\pi}{4}\), the values are \(x = r \cos \frac{\pi}{4}\) and \(y = r \sin \frac{\pi}{4}\). Similarly, for \(r = 2\), we compute \(x = 2 \cos \theta\) and \(y = 2 \sin \theta\). This accurate transformation facilitates finding solutions and verifying intersection points efficiently. Understanding and mastering coordinate transformations is essential in subjects like physics, engineering, and computer graphics, where it's used extensively.