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Chapter 7: Problem 105

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.

Short Answer

Expert verified
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 by step solution

01

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.
02

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 \((0, -4)\).
03

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.
04

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.

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Most popular questions from this chapter

Determine whether each statement makes sense or does not make sense, and explain your reasoning. The proof of the formula for the product of two complex numbers in polar form uses the sum formulas for cosines and sines that I studied in the previous chapter.

Use a graphing utility to graph the polar equation. $$r=2 \cos \left(\theta-\frac{\pi}{4}\right)$$

What is the polar form of a complex number?

Use a graphing utility to graph the polar equation. $$r=4 \cos 6 \theta$$

Use a graphing utility to graph each butterfly curve. Experiment with the range setting, particularly \(\theta\) step, to produce a butterfly of the best possible quality. $$r=\sin ^{5} \theta+8 \sin \theta \cos ^{3} \theta$$
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