Problem 16 - Precalculus

Question

Convert each point to exact polar coordinates. Assume that \(0 \leq \theta<2 \pi.\) $$(0, \sqrt{2})$$

Step-by-Step Solution

Answer

Polar coordinates are \\(\left(\sqrt{2}, \frac{\pi}{2}\right)\\).

Step 1: Identify the Cartesian Coordinates

The given Cartesian coordinates are \(x = 0, y = \sqrt{2}\). These coordinates describe a point in the Cartesian plane.

Step 2: Calculate the Radius (r)

The radius in polar coordinates is the distance from the origin to the point. Use the formula: \[ r = \sqrt{x^2 + y^2} \]Substitute the known values: \[ r = \sqrt{0^2 + (\sqrt{2})^2} = \sqrt{2} \]So, \(r = \sqrt{2}\).

Step 3: Calculate the Angle (θ)

The angle in polar coordinates is measured from the positive x-axis to the point. Since \(x = 0\) and \(y = \sqrt{2}\), the point lies on the positive y-axis. The angle \(\theta \) for this point is \(\frac{\pi}{2}\) (90 degrees).

Step 4: Write the Polar Coordinates

Combine the results from previous steps to write the polar coordinates. The polar coordinates are expressed as \(r, \theta\). Therefore, the exact polar coordinates are: \(\left(\sqrt{2}, \frac{\pi}{2}\right)\).

Key Concepts

Cartesian coordinates

Cartesian coordinates are a way to pinpoint locations on a plane using a pair of numerical values. They are generally expressed in the form \((x, y)\) where:

• x is the horizontal position, indicating how far the point is along the x-axis (left or right).
• y is the vertical position, reflecting how far the point is up or down along the y-axis.

The Cartesian system is named after the mathematician René Descartes, who devised it to combine geometry and algebra.
The coordinates \((0, \sqrt{2})\) tell us that the point is on the y-axis because the x-value is zero. Specifically, the point is at the positive square root of two units above the origin.
This is a rectangular coordinate system fundamental for plotting points and graphing various equations.

radius calculation

The radius in polar coordinates determines how far every point is from a central reference point, called the origin. To convert Cartesian coordinates \((x, y)\) into polar form, you first need to find this radius.

• The formula for calculating the radius \(r\) is: \[ r = \sqrt{x^2 + y^2} \]
• The equation derives from the Pythagorean theorem, where \(x\) and \(y\) are the perpendicular legs of a right triangle and \(r\) is the hypotenuse.

In our example, the coordinates were \((0, \sqrt{2})\), which simplifies the calculation to \(r = \sqrt{2}\).
This shows that the distance from the origin to the point on the Cartesian plane is the square root of 2 units.

angle calculation

Angle calculation is crucial for expressing Cartesian points in polar coordinates. The angle \(\theta\) tells us the direction from the positive x-axis to the line connecting the origin and the point.
Here’s how you determine it:

• If the point is on the x-axis, the angle is either 0 or \(\pi\) depending on the direction: right or left.
• If the point lies directly on the y-axis, like our point \((0, \sqrt{2})\), the angle is either \(\frac{\pi}{2}\) (upwards) or \(\frac{3\pi}{2}\) (downwards).
• Other angles must be calculated using trigonometric functions like \(\tan^{-1}(\frac{y}{x})\).

For this problem, the point is located on the positive y-axis, aligning exactly with \(\theta = \frac{\pi}{2}\), or 90 degrees. This angle reflects the direction towards which the radius extends.

coordinate conversion

The final step in translating from Cartesian to polar coordinates is coordinate conversion. Here's a clear breakdown:

• Combine the radius \(r\) and the angle \(\theta\) to express the location in polar form \((r, \theta)\).
• Polar coordinates focus on both the distance from the origin and the direction, offering a different perspective from Cartesian representations.

Converting \((0, \sqrt{2})\) to polar form, we determined \(r = \sqrt{2}\) and \(\theta = \frac{\pi}{2}\).
Thus, the polar coordinates are represented as \((\sqrt{2}, \frac{\pi}{2})\).
This transformation allows us to describe the point using polar notation, which is particularly useful in scenarios involving circular and rotational symmetries.