Problem 35 - Algebra and Trigonometry

Question

Convert the rectangular coordinates to polar coordinates with \(r>0\) and \(0 \leq \theta<2 \pi .\) $$ (-1,1) $$

Step-by-Step Solution

Answer

The polar coordinates are \(\left(\sqrt{2}, \frac{3\pi}{4}\right)\).

Step 1: Understand the Rectangular Coordinates

Rectangular coordinates are given as \((-1, 1)\), where \(x = -1\) and \(y = 1\). These coordinates describe a point in the Cartesian plane.

Step 2: Calculate the Radius \(r\)

The radius \(r\) in polar coordinates is calculated using the formula \(r = \sqrt{x^2 + y^2}\). Substitute the given \(x\) and \(y\) values: \[ r = \sqrt{(-1)^2 + 1^2} = \sqrt{1 + 1} = \sqrt{2} \approx 1.41 \].

Step 3: Calculate the Angle \(\theta\)

The angle \(\theta\) can be found using \(\theta = \tan^{-1}\left(\frac{y}{x}\right)\). For \((x, y) = (-1, 1)\), this becomes \[ \theta = \tan^{-1}\left(\frac{1}{-1}\right) = \tan^{-1}(-1) \]. Since the point lies in the second quadrant, we add \(\pi\) to adjust the angle, resulting in \[ \theta = \tan^{-1}(-1) + \pi \approx \frac{3\pi}{4} \].

Step 4: Combine \(r\) and \(\theta\) into Polar Coordinates

The polar coordinates are represented as \((r, \theta)\). Substituting the values found: \(r = \sqrt{2}\), \(\theta = \frac{3\pi}{4}\), the polar coordinates are \[ \left(\sqrt{2}, \frac{3\pi}{4}\right) \].

Key Concepts

Rectangular Coordinates

Rectangular coordinates are a way to represent points on a two-dimensional plane using a pair of numbers. These numbers are commonly denoted as \(x\) and \(y\), corresponding to horizontal and vertical positions, respectively. It is often referred to as the Cartesian coordinate system, named after the French mathematician René Descartes.

• The \(x\) coordinate shows how far left or right a point is from the origin (0, 0).
• The \(y\) coordinate indicates the point's vertical position, showing how far up or down it is from the origin.

These coordinates allow mathematicians and scientists to map out locations precisely on what is often called the Cartesian plane.

Cartesian Plane

The Cartesian plane is a flat surface filled with horizontal and vertical lines that form a grid. This grid is created by two perpendicular lines, known as the x-axis and y-axis, which intersect at the origin.The plane is divided into four sections or quadrants:

• The first quadrant is where both \(x\) and \(y\) coordinates are positive.
• The second quadrant has a negative \(x\) and a positive \(y\) coordinate.
• The third quadrant shows both coordinates as negative.
• The fourth quadrant is where \(x\) is positive, and \(y\) is negative.

Each point on the plane can be uniquely defined using a pair of rectangular coordinates.

Radius Calculation

When converting from rectangular to polar coordinates, the first step is to find the radius, \(r\), which measures the distance from the point to the origin on the Cartesian plane.The formula for radius calculation is \( r = \sqrt{x^2 + y^2} \).

• This formula is derived from the Pythagorean theorem, as the radius forms the hypotenuse of a right triangle in the plane.
• Substituting the given coordinates \((-1, 1)\), the calculation is: \[ r = \sqrt{(-1)^2 + 1^2} = \sqrt{1 + 1} = \sqrt{2} \].
• This gives \( r \approx 1.41\), which is the length of the diagonal line from the origin to the point \((-1, 1)\).

Angle Calculation

To find the angle \(\theta\) in polar coordinates, the tangent function is often used. This process determines the angle formed with the positive x-axis in the Cartesian plane.Calculate the angle using the formula: \( \theta = \tan^{-1}\left(\frac{y}{x}\right) \).

• For \((-1, 1)\), this becomes \( \theta = \tan^{-1}\left(\frac{1}{-1}\right) = \tan^{-1}(-1) \).
• Because the point is in the second quadrant, add \( \pi \) to shift the angle into the correct quadrant, resulting in \[ \theta = \tan^{-1}(-1) + \pi = \frac{3\pi}{4} \].
• This adjustment ensures that the angle measurement matches the position of the point on the plane.

This calculation ensures a comprehensive transition from rectangular to polar coordinates.