91Ó°ÊÓ

In the Cartesian coordinate system, points in a plane are identified using an ordered pair of numbers called coordinates. These coordinates are written as \((x, y)\), where:

• \textbf{x}: the horizontal distance from the origin.
• \textbf{y}: the vertical distance from the origin.

For the point \((-\sqrt{2}, \sqrt{2})\), \text{x = -\sqrt{2}}\ and \text{y = \sqrt{2}}\. This places the point in the second quadrant because:

• x is negative (to the left of the origin).
• y is positive (above the origin).

Understanding these coordinates is the first step towards efficiently converting them to polar coordinates.

Polar coordinates offer a different way to describe a point in a plane using two values: the radius \( r \) and the angle \( \theta \). The polar coordinates for a point are written as \( (r, \theta) \), where:

• \textbf{r}: the distance from the origin to the point.
• \textbf{\theta}: the angle measured counterclockwise from the positive x-axis to the line segment from the origin to the point.

For the point \((-\sqrt{2}, \sqrt{2})\):
1. Compute \( r \) using the formula: \[ r = \sqrt{x^2 + y^2} \] Substituting the values, \[r = \sqrt{ (-\sqrt{2})^2 + (\sqrt{2})^2 } = 2 \]
2. Compute \(\theta\) by finding the angle whose tangent is the ratio of y to x:
\[tan(\theta) = \frac{y}{x} = \frac{\sqrt{2}}{-\sqrt{2}} = -1 \] Since the point is in the second quadrant, \( \theta = 135^{\text{\degree}} \). So, one set of polar coordinates is \( (2, 135^{\text{\degree}}) \).
Another equivalent angle option:
3. To find another angle, we can add 360° or subtract 360° to keep within one cycle: \ [{0^{\text{\degree}} \leq \theta< 360^{\text{\degree}}}] For example:
Adding: \( 135^{\text{\degree}} + 360^{\text{\degree}} = 495^{\text{\degree}} \)
Subtracting: \( 135^{\text{\degree}} - 360^{\text{\degree}} = -225^{\text{\degree}} = 135^{\text{\degree}} \)
Using a negative radius with adjusted angle: \( (-2, 315^{\text{\degree}}) \)
Therefore, another set of polar coordinates is \( (-2, 315^{\text{\degree}}) \).

Converting a point from Cartesian to polar coordinates involves translating the rectangular coordinates into a form that uses distance and angle. This transformation is crucial for understanding relationships between points in different coordinate systems.
Here’s a quick guide:

• Distance (radial distance or r): Calculated using the Pythagorean theorem. From \((x, y)\) to \(r\), use:
  \[ r = \sqrt{x^2 + y^2} \]
  
• Angle (θ): Found using trigonometric relationships, specifically the tangent function:
  \[ \theta = \tan^{-1} \( \frac{y}{x} \) \] To place the angle correctly in the appropriate quadrant, consider the signs of x and y. Knowing the quadrant helps determine the correct direction and angle from the x-axis.

With these steps, any rectangular coordinate can be transformed into polar coordinates, providing a practical application for analyzing points in fields like physics, engineering, and robotics.