Problem 52 - Precalculus with Limits

Question

In Exercises 37-54, a point in rectangular coordinates is given. Convert the point to polar coordinates. \(\left(6, 2\right)\)

Step-by-Step Solution

Answer

The polar coordinates of the point \((6,2)\) are \((2\sqrt{10} , arctan(1/3))\).

Step 1: Calculate the radius

The radius \(r\) in polar coordinates is the distance from the origin to the point \((x,y)\), which can be found using the Pythagorean theorem. This gives \(r = \sqrt{x^2 + y^2}\). For the point \((6,2)\), we have \(r = \sqrt{6^2 + 2^2} = \sqrt{40} = 2\sqrt{10}\).

Step 2: Calculate the angle

The angle \(\theta\) is found by using the tangent function. Since \(tan(\theta) = y/x\) and our point is \((6,2)\), we find that \(\theta = arctan(y/x) = arctan(2/6) = arctan(1/3)\). For positive x and y, this gives the principal value of \(\theta\). But remember that there could be other valid values for \(\theta\) depending on what quadrant the point lies in.

Step 3: Combine radius and angle in polar coordinates

We now have both the radius and the angle, which are components of the polar coordinate. So we can directly write out the polar coordinate form as \( (2\sqrt{10} , arctan(1/3)) \).

Key Concepts

Rectangular to Polar Coordinates

When we come across a set of coordinates in a rectangular (also known as Cartesian) system, we might need to convert them into polar coordinates for various mathematical applications. The process of converting rectangular coordinates to polar coordinates involves finding the radius and the angle from the origin to the point.

For a given point with coordinates \( (x, y) \), the radius \( r \) can be determined using the distance formula derived from the Pythagorean theorem: \( r = \sqrt{x^2 + y^2} \). The angle \( \theta \) with the positive x-axis is calculated using the inverse tangent function: \( \theta = arctan(\frac{y}{x}) \). It's important to note the quadrant in which the point is located, as this affects the angle's value.

Let's consider an example with the point \( (6, 2) \). First, we calculate the radius: \( r = \sqrt{6^2 + 2^2} = \sqrt{40} = 2\sqrt{10} \). Next, we find the angle: \( \theta = arctan(\frac{2}{6}) = arctan(\frac{1}{3}) \). Knowing the x-coordinate is positive, and so is the y-coordinate, it places the point in the first quadrant, where the angle's principal value is used. Thus, the corresponding polar coordinates are \( (2\sqrt{10}, arctan(\frac{1}{3})) \).

Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in geometry that ties together the lengths of the sides of a right triangle. Stated simply, for a right triangle with legs \( a \) and \( b \) and hypotenuse \( c \) , the theorem explains that \( a^2 + b^2 = c^2 \).

When converting rectangular coordinates \( (x, y) \) to polar coordinates, we use a version of this theorem to find the radius. Imagine a right triangle formed by the x-axis, the vertical line through the given point, and the line segment connecting the origin to the point. Here, the legs of the triangle are the x and y coordinates, and the hypotenuse is the radius \( r \). Therefore, according to the Pythagorean theorem, \( r = \sqrt{x^2 + y^2} \).

This theorem is not only important for coordinate conversions but also serves as the foundation of many concepts within trigonometry and Euclidean geometry. It's the gateway to understanding distances and relationships within the Cartesian plane.

Polar Coordinates System

The polar coordinate system is an alternative to the rectangular coordinate system for representing points in a plane. Rather than using two perpendicular axes to mark the x and y positions, we use an angle and a distance from the origin.

In this system, a point is described by \( (r, \theta) \) where \( r \) stands for the radius or distance from the origin to the point, and \( \theta \) denotes the angle measured from the positive x-axis to the line segment connecting the origin to the point. The angle can be measured in degrees or radians and may require an adjustment based on which of the four quadrants the point resides in.

The polar coordinate system is particularly useful in scenarios where the relationship between two points is more easily understood in terms of angles and distances, such as in certain types of mathematical modeling, physics problems (such as those involving circular motion), and engineering fields. It emphasizes the radial symmetry of problems which might be cumbersome to analyze in a rectangular system.