Problem 10 - Precalculus: Functions and Graphs

Question

Change the rectangular coordinates to polar coordinates with \(r>0\) and \(0 \leq \theta \leq 2 \pi\). (a) \((3 \sqrt{3}, 3)\) (b) \((2,-2)\)

Step-by-Step Solution

Answer

(a) \((6, \frac{\pi}{6})\); (b) \((2\sqrt{2}, \frac{7\pi}{4})\)."

Step 1: Identify the Rectangular Coordinates

For part (a), the rectangular coordinates are \((3\sqrt{3}, 3)\). For part (b), the coordinates are \((2, -2)\).

Step 2: Calculate the Radius (r)

The formula for the radius \(r\) in polar coordinates is \(r = \sqrt{x^2 + y^2}\).- For part (a):\[ r = \sqrt{(3\sqrt{3})^2 + 3^2} = \sqrt{27 + 9} = \sqrt{36} = 6.\]- For part (b): \[ r = \sqrt{2^2 + (-2)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2}.\]

Step 3: Calculate the Angle (θ)

The formula for the angle \(\theta\) in polar coordinates is \(\theta = \tan^{-1}(\frac{y}{x})\).- For part (a): \[ \theta = \tan^{-1}\left(\frac{3}{3\sqrt{3}}\right) = \tan^{-1}\left(\frac{1}{\sqrt{3}}\right) = \frac{\pi}{6}. \]- For part (b): \[ \theta = \tan^{-1}\left(\frac{-2}{2}\right) = \tan^{-1}(-1) = \frac{7\pi}{4}. \] (Since the point is in the fourth quadrant, we add \(2\pi\) to \(-\frac{\pi}{4}\) to adjust it to the correct interval.)

Step 4: Write the Polar Coordinates

Now that we have both \(r\) and \(\theta\), we can write the polar coordinates.- For part (a): The polar coordinates are \((6, \frac{\pi}{6})\).- For part (b): The polar coordinates are \((2\sqrt{2}, \frac{7\pi}{4})\).

Key Concepts

Rectangular Coordinates

In mathematics, rectangular coordinates are a system that defines the position of a point in a plane using two values. These values, often represented as \((x, y)\), describe a point's location based on distance along two perpendicular axes: the horizontal x-axis and the vertical y-axis. This system is also known as the Cartesian coordinate system. By visualizing its grid-like structure, you can easily locate points on a plane.

Rectangular coordinates are central to graphing and algebra, always assisting in identifying precise positions and analyzing geometric shapes. A positive x-value indicates movement to the right from the origin, while a positive y-value indicates upward movement. Conversely, negative values imply movement left or downward.

Understanding rectangular coordinates is essential when transforming points into other systems, such as polar coordinates, as it provides the starting values needed for conversion.

Radius Calculation

In polar coordinates, the radius \(r\) represents the distance of a point from the origin. To compute the radius, we use the distance formula derived from the Pythagorean theorem:

• \(r = \sqrt{x^2 + y^2}\)

This formula measures how far a point in the Cartesian plane (rectangular coordinates) is from the center, which is always at the origin (0,0).

Consider a point \((x, y)\):

• Square both the x and y values.
• Add these squares together.
• Finally, take the square root of the sum to find \(r\).

Calculating the radius is a straightforward process that establishes one part of the polar coordinate system. It's important to ensure \(r\) is always positive, as it simply measures distance.

Angle Calculation

The angle \(\theta\) in polar coordinates determines a point's direction from the positive x-axis. To find this angle, we use the inverse tangent function:

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

The angle \(\theta\) is measured in radians and typically lies within the interval \(0 \leq \theta < 2\pi\).

When calculating \(\theta\), you'll:

• Divide the y-coordinate by the x-coordinate.
• Use the arctangent function to find the angle \(\theta\).

It is essential to consider the quadrant in which the point \((x, y)\) lies. For example, if a point is in the fourth quadrant, as in the provided example with coordinates \((2,-2)\), you'll need to adjust \(\theta\) to ensure it falls within the correct range, often done by adding \(2\pi\) when necessary.

Coordinate Transformation

Coordinate transformation involves converting a point from one coordinate system to another, such as from rectangular to polar coordinates. This transformation helps in solving problems in various fields, including physics and engineering.

When performing this transformation:

• Calculate the radius \(r\) using \(\sqrt{x^2 + y^2}\).
• Find the angle \(\theta\) using \(\tan^{-1}\left(\frac{y}{x}\right)\) and adjust for the correct quadrant.
• The polar coordinates are expressed as \((r, \theta)\).

Transformations are essential for analyzing problems where circular or radial symmetry is more apparent in polar coordinates compared to rectangular coordinates. Understanding these transformations enhances one's ability to visualize and solve complex problems.