Problem 5 - Precalculus

Question

onvert the given rectangular coordinates to polar coordinates. Express your answers in such a way that ris nonnegative and \(0 \leq \theta<2 \pi\) $$(-1,-1)$$

Step-by-Step Solution

Answer

The polar coordinates are \((\sqrt{2}, \frac{5\pi}{4})\).

Step 1: Identify Components

In the given rectangular coordinates (-1, -1), identify the values of the x and y components. Here, the x-coordinate is -1 and the y-coordinate is -1.

Step 2: Calculate Radius

The radius \( r \) in polar coordinates is calculated using the formula \( r = \sqrt{x^2 + y^2} \). Substitute \( x = -1 \) and \( y = -1 \): \[ r = \sqrt{(-1)^2 + (-1)^2} = \sqrt{1 + 1} = \sqrt{2} \] Thus, \( r = \sqrt{2} \).

Step 3: Calculate Initial Theta

The angle \( \theta \) is given by \( \theta = \arctan\left(\frac{y}{x}\right) \). For \( x = -1 \) and \( y = -1 \): \[ \theta = \arctan\left(\frac{-1}{-1}\right) = \arctan(1) = \frac{\pi}{4} \] However, as the point (-1, -1) is in the third quadrant, adjust \( \theta \) by adding \( \pi \) to it. \[ \theta = \pi + \frac{\pi}{4} = \frac{5\pi}{4} \] Thus, \( \theta = \frac{5\pi}{4} \).

Step 4: Verify Theta Range

Ensure the calculated \( \theta \) is within the range \( 0 \leq \theta < 2\pi \). Given \( \theta = \frac{5\pi}{4} \), it is indeed within this range.

Key Concepts

Rectangular Coordinates

In the realm of mathematics, rectangular coordinates, also known as Cartesian coordinates, are used to precisely locate a point in a two-dimensional plane. These coordinates consist of two values, usually denoted as \( x \) and \( y \), which represent positions on the horizontal (x-axis) and vertical (y-axis) axes, respectively.

They give a simple numeric description of a point's location, making them very intuitive and widely used, especially in real life and in plotting graphs.

• The coordinate pair takes the form \((x, y)\).
• The origin, where both coordinates are zero, is denoted as \((0, 0)\).
• In our example, the rectangular coordinates given are \((-1, -1)\), indicating a point in the third quadrant.

Understanding rectangular coordinates is foundational for transitioning into other coordinate systems like polar coordinates.

Radius Calculation

When converting rectangular coordinates to polar coordinates, calculating the radius \( r \) is an essential step. The radius is the distance from the origin \((0, 0)\) to the point in the plane. This is mathematically achieved using the Pythagorean theorem.

To compute the radius \( r \), use the formula: \[ r = \sqrt{x^2 + y^2} \]
In this formula, \( x \) and \( y \) are the rectangular coordinates.

• Insert the corresponding values: for \((x, y) = (-1, -1)\), we substitute \( x = -1 \) and \( y = -1 \).
• This yields \( r = \sqrt{(-1)^2 + (-1)^2} = \sqrt{1 + 1} = \sqrt{2} \).
• Hence, the calculated radius is \( \sqrt{2} \).

The radius represents how far the point is from the origin, essential for plotting polar coordinates.

Angle Calculation

Determining the angle \( \theta \) for polar coordinates is another crucial step, indicating the direction from the origin to the point.

This is calculated using the formula: \[ \theta = \arctan\left(\frac{y}{x}\right) \]
For \((-1, -1)\), you'll set \( y = -1 \) and \( x = -1 \) giving:

• \( \theta = \arctan\left(\frac{-1}{-1}\right) = \arctan(1) = \frac{\pi}{4} \).
• However, since the point is located in the third quadrant, we need to add \( \pi \) to adjust the calculated angle.
• Thus, \( \theta = \pi + \frac{\pi}{4} = \frac{5\pi}{4} \).

This adjustment ensures the angle accurately reflects the point's position in the Cartesian plane, aligning it with the polar system.

Coordinate Conversion

Coordinate conversion involves transforming a point from one coordinate system to another, in this case from rectangular to polar coordinates. The conversion is completed by combining both the radius and angle calculations.

To fully convert, one must correctly calculate both the radius \( r = \sqrt{2} \) and ensured angle \( \theta = \frac{5\pi}{4} \) for the point \((-1, -1)\).

• Start by calculating the radius using the formula \( r = \sqrt{x^2 + y^2} \). In our example, this was \( \sqrt{2} \).
• Find the angle \( \theta \) using \( \arctan\left(\frac{y}{x}\right) \), adjusting as needed for the quadrant, resulting in \( \frac{5\pi}{4} \).
• Therefore, the polar coordinates can be expressed as \((r, \theta) = (\sqrt{2}, \frac{5\pi}{4})\).

This conversion is important for applications requiring angle and distance from a central point, useful in physics, engineering, and graphics.