Problem 22 - Precalculus with Limits

Question

In Exercises 19-28, a point in polar coordinates is given. Convert the point to rectangular coordinates. \(\left(0, -\pi\right)\)

Step-by-Step Solution

Answer

The rectangular coordinates for the given polar coordinates (0,-π) are (0,0).

Step 1: Identifying the given point and the conversion formulas

The point given in polar coordinates is (0,-π). The standard formulas to convert polar coordinates (r,θ) to rectangular coordinates (x, y) are as follows: x=r*cos(θ) and y=r*sin(θ).

Step 2: Applying the conversion formulas

Now, we substitute r=0 and θ=-π into the conversion formulas. For the x-coordinates, x=0*cos(-π) which gives x=0. Similarly, for y-coordinates, y=0*sin(-π), which also gives y=0.

Step 3: Writing the final answer

So, the rectangular coordinates for the given polar coordinates (0,-π) are (0,0).

Key Concepts

Polar Coordinates

Polar coordinates are a way of representing points in a plane using a distance and an angle. Instead of the regular Cartesian way of measuring from two perpendicular axes, polar coordinates measure from a central point, known as the pole (similar to the origin in rectangular coordinates).

Here's how polar coordinates are structured:

• **Distance (r)**: This is the radial distance from the pole to the point. It is the first coordinate in a polar pair, denoted as \( r \).
• **Angle (θ)**: This is the measure of the angle formed with the positive x-axis, moving counterclockwise. This angle is the second coordinate, denoted as \( \theta \). It is typically measured in radians or degrees.

Both these components (\( r, \theta \)) work together to pinpoint a particular location on the plane.

An interesting feature of polar coordinates is that there can be multiple pairs representing the same point, due to the periodic nature of angles and the allowance for negative distances.

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, form the foundation of most graphing and are used to locate points based on horizontal and vertical lines. In this system, each point in a plane is described by a pair of numerical values that correlate with how far along each axis the point is positioned.

A typical rectangular coordinate is represented as (x, y), where:

• **x-coordinate**: This number tells you how far to move horizontally from the origin. A positive value indicates a rightward movement, and a negative value indicates a leftward move.
• **y-coordinate**: This value indicates how far to move vertically from the origin. Positive values move you upward, whereas negative values move you downward.

These coordinates provide a clear and easy method to map exact positions on a two-dimensional plane, making them very useful for graphing functions and geometry.

Coordinate Conversion

Coordinate conversion is the process of changing from one coordinate system to another, such as from polar coordinates to rectangular coordinates. This is often necessary because different scenarios or problems might be easier to solve or visualize in one type of coordinates or the other.

To convert from polar coordinates (\( r, \theta \)) to rectangular coordinates (x, y), we use trigonometric functions:

• The formula for the x-coordinate is: \(
  x = r \cdot \cos(\theta) \)
  
• The formula for the y-coordinate is: \(
  y = r \cdot \sin(\theta) \)
  

These formulas use the cosine and sine of the angle to transform the 'radius' and angle format of polar coordinates into the 'horizontal' and 'vertical' distances of rectangular coordinates. This technique is very practical in physics and engineering, where circular or rotational systems are involved.

Trigonometric Functions

Trigonometric functions are essential in the conversion of coordinate systems because they relate angles to side lengths in right triangles. They are broadly used to understand periodic phenomena, and in this context, they help in transitions between polar and rectangular coordinate systems.

Key trigonometric functions that are especially important here include:

• **Cosine (\( \cos(\theta) \))** - This function returns the horizontal coordinate of a unit circle segment. In coordinate conversion, it helps determine the x-component in rectangular coordinates.
• **Sine (\( \sin(\theta) \))** - This function returns the vertical coordinate of a unit circle segment, essential for finding the y-component in rectangular coordinates.

Both these functions rely on geometric principles from the unit circle, where the circle's radius is 1. As \( \theta \) changes, the (\( \cos(\theta), \sin(\theta) \)) values illustrate the point's movement around the circle, providing the basis for coordinate conversion through multiplication by \( r \), the given radius in polar coordinates.