Problem 62 - Algebra and Trigonometry

Question

The rectangular coordinates of a point are given. Find polar coordinates for each point. $$ (0,-2) $$

Step-by-Step Solution

Answer

(2, -π/2)

Step 1 - Identify the given rectangular coordinates

The given rectangular coordinates are (0, -2). This means that the point is located at x = 0 and y = -2.

Step 2 - Calculate the radius (r)

The radius in polar coordinates is given by the formula \( r = \sqrt{x^2 + y^2} \). Plugging in the values, we get \( r = \sqrt{0^2 + (-2)^2} = \sqrt{4} = 2 \). So, the radius is r = 2.

Step 3 - Calculate the angle (θ)

The angle in polar coordinates, θ, is calculated using the formula \( θ = \arctan \left( \frac{y}{x} \right) \). In this case, \( x = 0 \), so we have to determine θ directly. Since the point is on the negative y-axis, θ is \(-\pi/2\) radians (or -90 degrees).

Step 4 - Write the polar coordinates

Combining the radius and the angle calculated, the polar coordinates of the point (0, -2) are (2, -π/2).

Key Concepts

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, are a method of locating points on a plane using two numbers, typically written as \(x, y\). Each point on the plane is defined by its horizontal (x) and vertical (y) distances from the origin (0,0). In the given exercise, the rectangular coordinates are (0, -2).
This means that the point lies directly on the y-axis, 2 units below the origin. Understanding this positional relationship is essential before converting these coordinates to polar coordinates.

Radius Calculation

The radius (r) in polar coordinates is a measure of distance from the origin to a point. We use the Pythagorean Theorem to find this distance. The formula for the radius is: \[ r = \sqrt{x^2 + y^2} \] Here, x and y are the rectangular coordinates.
Plugging in the given coordinates (0, -2):
\[ r = \sqrt{0^2 + (-2)^2} = \sqrt{4} = 2 \] So, the radius is 2. This value tells us that the point is 2 units away from the origin, regardless of direction.

Angle Determination

In polar coordinates, the angle (θ) determines the direction from the origin to the point. This angle is usually measured in radians. To determine θ, we use the arctangent (or inverse tangent) of the ratio between y and x: \[ θ = \arctan \left( \frac{y}{x} \right) \] However, when x is 0, we cannot directly compute the ratio.
Instead, we need to look at the point's position relative to the coordinate axes. For the point (0, -2), it lies on the negative y-axis, which corresponds to an angle of \(-\frac{\pi}{2}\) radians (or -90 degrees).

Combining the radius and the angle, the polar coordinates of the point (0, -2) are (2, \(-\pi/2\)).