91Ó°ÊÓ

When we talk about Cartesian coordinates, we're diving into the world of a two-dimensional plane most students are familiar with. Cartesian coordinates are expressed as an ordered pair

• Where the first element represents the position along the horizontal axis (x-axis)
• And the second element represents the position along the vertical axis (y-axis)

In our given exercise, the Cartesian coordinates are \( (x, y) = (\sqrt{6}, -\sqrt{2}) \).This is essentially a way to describe where a point lies on a flat surface or graph.

Thus each point on this plane is defined by these two numerical values, making it easier to locate it precisely on the grid.

To transition from Cartesian to polar coordinates, we first need to calculate the radius. The radius, in the context of polar coordinates, refers to the direct distance from the origin to the point. It gives us a clearer sense of how far away the point is in terms of a circle centered at the origin.

To find this radius, we apply the Pythagorean Theorem in a formula: \[ r = \sqrt{x^2 + y^2} \]Consequently, inputting \( x = \sqrt{6} \) and \( y = -\sqrt{2} \),we find: \[ r = \sqrt{(\sqrt{6})^2 + (-\sqrt{2})^2} = \sqrt{6 + 2} = \sqrt{8} = 2\sqrt{2} \]By calculating this, we essentially gain an understanding of the point’s absolute distance from the origin, crucial for converting into a circular or polar perspective.

Calculating the angle \( \theta \) is a key step in converting to polar coordinates. The angle tells us the direction from the origin to the point. To find this angle, we use the arctangent function which is ideal for finding angles based on \( y/x \) ratios.

The formula is:\[ \theta = \tan^{-1}\left(\frac{y}{x}\right) \]Substituting the values from our problem:\[ \theta = \tan^{-1}\left(\frac{-\sqrt{2}}{\sqrt{6}}\right) \]This simplifies to \(-\frac{1}{\sqrt{3}}\),where\[ \theta = \tan^{-1}\left(-\frac{1}{\sqrt{3}}\right) \]This results in an angle of :\(-\frac{\pi}{6}\)However, because polar coordinates generally express angles in a range from \([0, 2\pi)\),we convert this angle to a positive measure: \(11\frac{\pi}{6}\).

The ultimate goal of our task is to convert from Cartesian to polar coordinates. This process involves transitioning from x and y positions on a grid to a circular framework defined by radius and angle.

• The radius \(r\) is calculated as \( 2\sqrt{2} \)
• The angle \(\theta\) is found to be \(11\frac{\pi}{6}\).

The resulting polar coordinates are then expressed as \((r, \theta) = (2\sqrt{2}, 11\frac{\pi}{6})\).

These coordinates tell us not just the distance from the origin, but also the specific direction relative to a standard position. Polar coordinates are often favored in contexts where direction and radius are more intuitive or relevant than the Cartesian method. This conversion provides a different perspective, spotlighting the versatility and depth of coordinate systems in mathematics.