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Chapter 6: Problem 50

In Exercises \(43-60,\) a point in rectangular coordinates is given. Convert the point to polar coordinates. $$(0,5)$$

Short Answer

Expert verified
The point (0,5) in rectangular coordinates corresponds to (5, \(\frac{\pi}{2}\)) in polar coordinates.

Step by step solution

01

Calculate r

Firstly, we calculate the radial coordinate r. The formula to find r is \(r = \sqrt{x^2 + y^2}\). Here, x = 0 and y = 5 so \(r= \sqrt{0^2 + 5^2} = \sqrt{25} = 5\)
02

Calculate 胃

To find the angular coordinate 胃, we generally use the inverse tangent function with y as the numerator and x as the denominator, i.e., \(胃 = \arctan(\frac{y}{x})\). Here, x = 0 and y = 5, so \(胃 = \arctan(\frac{5}{0})\). However, arctan is not defined when the denominator is 0, as we have in this case. Instead, knowing that we're on the positive y-axis, we set the angle 胃 to be either \(胃 = 90^{\circ}\) or \(胃 = \frac{\pi}{2}\) radians.
03

Write down the point in polar coordinates

The point in polar coordinates is then given by (r, 胃). From step 1 and 2, we have r = 5 and \(胃 = \frac{\pi}{2}\) So the polar coordinates are given by (5, \(\frac{\pi}{2}\))

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Rectangular Coordinates
Rectangular coordinates, also known as Cartesian coordinates, are a fundamental concept in mathematics. They are used to specify the position of a point in a two-dimensional plane. A point in rectangular coordinates is defined by an ordered pair \(x, y\). This means:
  • \(x\) is the horizontal position relative to the origin (0,0).
  • \(y\) is the vertical position relative to the origin.
The system is named after the French mathematician Ren茅 Descartes. Rectangular coordinates make it easier to perform algebraic calculations and transformations. Converting a point from rectangular to polar coordinates involves determining the point's distance from the origin and its angle relative to the positive x-axis. This exercise helps deepen your understanding of coordinate transformations and geometry.
Radial Coordinate
The radial coordinate, denoted by \(r\), is a crucial component of polar coordinates. Unlike the straightforward horizontal and vertical measurements of rectangular coordinates, the radial coordinate represents:
  • The distance of the point from the origin (0,0).
To calculate \(r\), you use the Pythagorean theorem, represented by the formula \(r = \sqrt{x^2 + y^2}\). This formula sets the length of the line (radius) from the origin to the point \(x,y\). In our example, with \(x = 0\) and \(y = 5\), the radial coordinate is 5. Understanding the radial coordinate is key to transitioning comfortably between rectangular and polar systems, making complex computations or transformations easier to handle.
Angular Coordinate
The angular coordinate in polar coordinates, denoted \(胃\), is the angle measurement from the positive x-axis to the line connecting the origin and the point. It provides the direction in which the radius \(r\) is pointing. This angle is typically measured in radians or degrees.
  • For a point on the positive y-axis, the angle is \(90^{\circ}\) or \(\frac{\pi}{2}\) radians.
Calculating \(胃\) can usually be done using inverse trigonometric functions. However, special cases arise when the point lies along one of the axes (like \(x = 0\)), requiring direct angle assignment. Mastery of this concept aids in visualizing and solving geometrical problems involving angles and directions.
Inverse Tangent Function
The inverse tangent function, often written as \( an^{-1}\) or \( ext{arctan}\), is used to determine the angle \(胃\) from a given slope, which is the ratio of the change in \(y\) to \(x\) (i.e., \(\frac{y}{x}\)). It is a fundamental tool in converting coordinates from rectangular to polar form.
  • It returns the angle in radians or degrees.
  • Simplifies the trigonometric calculation of angle given a slope.
A drawback occurs when \(x = 0\), as division by zero is undefined. In such scenarios, the standard arctan function fails, requiring intuitive angle interpretation based on the point's position. Becoming adept with the inverse tangent function strengthens your skills in trigonometry and analytical geometry.

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