Problem 105 - Algebra and Trigonometry

Question

Determine whether each statement makes sense or does not make sense, and explain your reasoning. After plotting the point with rectangular coordinates \((0,-4),\) I found polar coordinates without having to show any work.

Step-by-Step Solution

Answer

The statement partially makes sense. We can tell that the radial coordinate is 4 directly by its distance from the origin. However, determining the angular coordinate requires knowledge of the relationship between the points on the Cartesian plane and the angle measures in a polar coordinate system. Here it is -90 degrees or -\(\pi/2\) radians as the point lies on the negative y-axis.

Step 1: Understand Coordinate Systems

In the planar coordinate systems, there are two types: rectangular (or Cartesian) which uses two perpendicular axes to describe a point in the plane by an ordered pair of numerical coordinates. Polar Coordinate System uses a point's distance and angle from a fixed point. The fixed point is called the pole (or origin), and the ray from the pole in the fixed direction is the polar axis.

Step 2: Conversion from Rectangular to Polar Coordinates

The conversion from rectangular coordinates \((x, y)\) to polar coordinates \((r, \Theta)\) is done using the following equations : \(r = \sqrt{x^2 + y^2}\) and \(\Theta = \arctan(y/x)\) (this gives the angle in radians). In this exercise, the point has rectangular coordinates of \((0, -4)\).

Step 3: Apply The Conversion To The Given Point

Substitute the given coordinates into the conversion formulas: \(r = \sqrt{0^2 + (-4)^2} = 4\) and the angle \(\Theta = \arctan((-4)/0)\) is undefined as division by zero is undefined. However, in this case, the point lies on the negative y-axis. So, by convention, we can take \(\Theta = -90^o\) or \(-\(\pi/2\) radians.

Step 4: Check the Statement

The original statement claimed that the polar coordinates could be found without any calculations. From our steps above, we can see this is partially true. The radial coordinate \(r=4\) can be found easily as it is the distance from origin. However, finding the angular coordinate \(\Theta\) needed an understanding of the position of point in the plane.