Problem 27 - Contemporary Precalculus

Question

Convert the rectangular coordinates to polar coordinates. $$(2,4)$$

Step-by-Step Solution

Answer

Question: Convert the rectangular coordinates (2, 4) to polar coordinates. Answer: The rectangular coordinates (2, 4) can be converted to polar coordinates as approximately $(\sqrt{20}, 63.43^{\circ})$.

Step 1: Calculate r, the distance from the origin to the point

Using the Pythagorean theorem, we can find the distance r between the origin (0,0) and the point (2,4) as follows: $$r = \sqrt{(x-0)^2 + (y-0)^2}$$ Plugging in the coordinates: $$r = \sqrt{(2-0)^2 + (4-0)^2} = \sqrt{4 + 16} = \sqrt{20}$$

Step 2: Calculate θ, the angle between the positive x-axis and the vector

To find the angle θ, let's use the inverse tangent function: $$\theta = \tan^{-1}\left(\frac{y}{x}\right)$$ Plugging in the coordinates: $$\theta = \tan^{-1}\left(\frac{4}{2}\right) = \tan^{-1}(2)$$ To get the angle in degrees, let's convert the result from radians to degrees: $$\theta = \tan^{-1}(2) \cdot \frac{180^{\circ}}{\pi} \approx 63.43^{\circ}$$

Step 3: Write the result in polar coordinates

Now that we found the distance r and the angle θ, we can write the polar coordinates as: $$(r, \theta) = \left(\sqrt{20}, \tan^{-1}(2)\right)$$ or in degrees: $$(r, \theta) \approx \left(\sqrt{20}, 63.43^{\circ}\right)$$ So, the rectangular coordinates (2,4) can be converted to polar coordinates as approximately $$(\sqrt{20}, 63.43^{\circ})$$.

Key Concepts

Rectangular Coordinates

Rectangular coordinates are a way to represent the position of a point in a plane using two perpendicular lines, commonly known as the x-axis and y-axis. In this system, any point is described by an ordered pair \((x, y)\), where \(x\) indicates the horizontal position and \(y\) the vertical position.

The rectangular coordinate system is also referred to as the Cartesian coordinate system, named after the French mathematician René Descartes. This method is widely used due to its straightforward representation and ease of understanding geometric properties visually.

• The \(x\) value signifies the position along the horizontal axis, left or right of the origin (0, 0).
• The \(y\) value denotes the position along the vertical axis, above or below the origin.

To illustrate, the point (2, 4) means that it is located two units in the positive x direction and four units in the positive y direction from the origin.

Coordinate Conversion

Coordinate conversion is the process of transforming coordinates expressed in one system to another system, in this case, from rectangular to polar coordinates. This conversion is essential in various fields such as navigation, physics, and engineering, where analyzing spatial relationships is required in different forms.

In converting rectangular coordinates \((x, y)\) to polar coordinates \((r, \theta)\):

• The distance \(r\) from the origin (0,0) is calculated using the Pythagorean theorem.
• The angle \(\theta\) is determined using trigonometric functions, specifically the inverse tangent function.

This conversion allows us to describe the same point with a different perspective, highlighting its distance and direction from the origin rather than its horizontal and vertical displacement.

Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in geometry, relating the lengths of the sides of a right triangle. It states:

\[ c^2 = a^2 + b^2 \]

where \(c\) is the hypotenuse (the side opposite the right angle), and \(a\) and \(b\) are the other two sides. This theorem is incredibly useful for converting rectangular coordinates to polar coordinates, as it helps calculate the distance \(r\) from a point to the origin.

In this context, consider a point \((x, y)\).

• Think of \(x\) and \(y\) as legs of a right triangle with the origin at (0,0).
• The distance \(r\) is the hypotenuse, calculated as: \(r = \sqrt{x^2 + y^2}\).

Using this theorem simplifies the process of finding the magnitude or distance in polar coordinates, a key element in transforming from the rectangular format.

Inverse Tangent Function

The inverse tangent function, also known as the arctangent and denoted as \(\tan^{-1}\), is a trigonometric function used to find an angle when the tangent ratio is known. In the context of coordinate conversion, it is crucial for determining the angle \(\theta\) in polar coordinates.

To convert rectangular coordinates \((x, y)\) to polar coordinates, use the formula:

\[ \theta = \tan^{-1}\left(\frac{y}{x}\right) \]

This formula uses the \(y\) and \(x\) values to calculate the angle that the point \((x, y)\) makes with the positive x-axis.
This angle is often measured from the x-axis to the vector drawn from the origin to the point.

• It is important to note whether the angle is measured in radians or degrees.
• Often, additional conversion from radians to degrees is necessary, especially in practical applications that favor degree measurements.

Such transformations help relate spatial problems to angular measurements, vital for applications across multiple disciplines.