Problem 34 - Contemporary Precalculus

Question

Convert the rectangular coordinates to polar coordinates. $$(\sqrt{5}, \sqrt{10})$$

Step-by-Step Solution

Answer

Question: Convert the rectangular coordinates \((\sqrt{5}, \sqrt{10})\) to polar coordinates. Answer: The polar coordinates are \(\left(\sqrt{15}, \arctan\left(\sqrt{2}\right)\right)\).

Step 1: Calculate the radius r

To find the radius \(r\), we'll use the Pythagorean theorem. Since the rectangular coordinates are \((x,y) = (\sqrt{5}, \sqrt{10})\), we can write the equation for \(r\) as follows: $$r = \sqrt{x^2 + y^2} = \sqrt{(\sqrt{5})^2 + (\sqrt{10})^2}$$ Now, compute the radius: $$r = \sqrt{5+10} = \sqrt{15}$$

Step 2: Determine the angle θ

We can use the arctangent function to determine the angle \(\theta\). First, we need to find the ratio \(\frac{y}{x}\). In this case, \(x = \sqrt{5}\) and \(y = \sqrt{10}\). Calculate the ratio: $$\frac{y}{x} = \frac{\sqrt{10}}{\sqrt{5}} = \sqrt{2}$$ Now, find the angle by taking the arctangent of the ratio: $$\theta = \arctan\left(\sqrt{2}\right)$$

Step 3: Write the polar coordinates

Finally, write the polar coordinates as a pair \((r, \theta)\): $$(r, \theta) = \left(\sqrt{15}, \arctan\left(\sqrt{2}\right)\right)$$ Thus, the polar coordinates of the given rectangular coordinates are \(\left(\sqrt{15}, \arctan\left(\sqrt{2}\right)\right)\).

Key Concepts

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, are an essential concept in mathematics when it comes to identifying the position of a point on a plane. These coordinates are given as a pair \(x, y\), where \(x\) is the horizontal distance from the origin, and \(y\) is the vertical distance. This system is named after René Descartes and is widely used to specify locations in 2D space.
The primary idea is to divide the plane into four quadrants, where both the horizontal and vertical axes intersect at the origin (0,0). Rectangular coordinates are incredibly beneficial because they allow complex mathematical problems to be solved systematically. Their use extends beyond mathematics into fields like engineering and computer graphics, where describing locations precisely is crucial.

Pythagorean Theorem

The Pythagorean Theorem is a fundamental principle in geometry that relates the sides of a right triangle. This theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
In the context of converting rectangular coordinates to polar coordinates, this theorem plays a pivotal role. For a point \( (x, y) \), the theorem helps compute the distance from the origin, commonly known as the radial distance \(r\). The formula applied is:

• \( r = \sqrt{x^2 + y^2} \)

This equation makes it possible to describe any point in terms of its distance from the origin and its angle, facilitating the conversion into the polar coordinate system.

Arctangent Function

The arctangent function, often written as \(\tan^{-1}(x)\) or \( ext{arctan}(x)\), is an inverse trigonometric function used to find the angle whose tangent is given. In simpler terms, it computes the angle associated with a tangent value.
This function is particularly useful in coordinate conversion processes. When transitioning from rectangular to polar coordinates, the arctangent function helps find the angle \(\theta\) between the positive x-axis and the line from the origin to the point \( (x, y) \).
The formula to determine \( \theta \) is:

• \( \theta = \arctan\left( \frac{y}{x} \right) \)

Using \(\tan^{-1}\), one can easily determine the angle, which, along with the radius, completes the transformation to polar coordinates.

Coordinate Conversion

Coordinate conversion is the process of transforming one type of coordinate system into another. Understanding how to switch between rectangular coordinates \(x, y\) and polar coordinates \(r, \theta\) is crucial for solving numerous mathematical and real-world applications efficiently.
To convert from rectangular to polar coordinates, the following steps are generally followed:

• Calculate the radius: \( r = \sqrt{x^2 + y^2} \)
• Determine the angle: \( \theta = \arctan\left( \frac{y}{x} \right) \)

The resultant polar coordinates give a comprehensive way to express the location of a point based on its distance from the origin and the angle from the reference direction.
This conversion is not just crucial in theoretical mathematics but also finds application in physics, navigation, and computer graphics, where polar coordinates simplify the understanding and calculation of rotational dynamics and wave motions.