Problem 86 - Contemporary Precalculus

Question

Prove that the coordinate conversion formulas are valid when \(r < 0 .[\text {Hint: If } P \text { has coordinates }(x, y) \text { and }(r, \theta), \text { with } r < 0\) verify that the point \(Q\) with rectangular coordinates \((-x,-y)\) has polar coordinates \((-r, \theta) .\) since \(r < 0,-r\) is positive and the conversion formulas proved in the text apply to \(Q .\) For instance, \(-x=-r \cos \theta, \text { which implies that } x=r \cos \theta .]\)

Step-by-Step Solution

Answer

Question: Prove the coordinate conversion formulas when r < 0 for a given point P(x, y). Answer: To prove the coordinate conversion formulas for P(x, y) when r < 0, we can find a point Q(-x, -y) with polar coordinates (-r, θ) where -r is positive. By following the steps mentioned in the solution, we can show that the conversion formulas for the rectangular and polar coordinates of the point P work even when r < 0. This proves the validity of the coordinate conversion formulas for r < 0.

Step 1: Find the point Q, given P

Given the point P with rectangular coordinates (x, y) and polar coordinates (r, θ), we need to define a point Q such that Q has rectangular coordinates (-x, -y).

Step 2: Convert Q's coordinates from rectangular to polar

As Q is defined by rectangular coordinates (-x, -y), we can convert these coordinates into polar coordinates -r and θ by using the already proven conversion formulas for positions where r > 0. \(r = \sqrt{(-x)^2 + (-y)^2}\) and \(\theta = \arctan{\frac{-y}{-x}}\) We can simplify this as: \(-r = \sqrt{x^2 + y^2}\) and \(\theta = \arctan{\frac{y}{x}}\) Then, we find that Q has polar coordinates (-r, θ), as required.

Step 3: Show the polar conversion formulas are consistent for P

Now, let us show that the polar conversion formulas work for P when r < 0: Polar coordinates of P are (r, θ), and the rectangular coordinates are (x, y). Polar coordinates of Q are (-r, θ), and the rectangular coordinates are (-x, -y). Since conversion formulas are proven for the case when r > 0, we can apply them to the point Q: \(-x = - r \cos{\theta}\) and \(-y = - r \sin{\theta}\) Multiplying both sides of each equation by -1, we obtain: \(x = r \cos{\theta}\) and \(y = r \sin{\theta}\) These conversion formulas adhere to the relationships of the rectangular and polar coordinates of point P when r < 0. Therefore, we can conclude that the coordinate conversion formulas are valid even when r < 0.

Key Concepts

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, represent a point in a two-dimensional space with an ordered pair \( (x, y) \). This system uses two perpendicular linear axes that intersect at a point known as the origin. The x-coordinate indicates the position along the horizontal axis, while the y-coordinate reflects the position along the vertical axis.

To understand rectangular coordinates better, imagine a grid of horizontal and vertical lines, similar to a checkerboard. Each point on the grid is defined by a unique pair of numbers: one for the horizontal displacement from the origin, and one for the vertical displacement.

When working with negative values in this system, such as \( (-x, -y) \), we are referring to a point that is located in relation to the opposite direction of both axes from the origin. In the context of converting coordinates, knowing how to plot points and move around this grid is fundamental.

Polar Coordinates

Polar coordinates, on the other hand, offer a different perspective for locating a point in a plane. Instead of using perpendicular axes, the polar system is based on circles and angles. A point in polar coordinates is given by a pair \( (r, \theta) \), where \( r \) is the radial distance from the origin (or pole) and \( \theta \) is the angle formed with the positive x-axis, also known as the polar axis.

It's important to note that while \( r \) is usually taken as a positive value, negative values are also meaningful. In such cases, \( r < 0 \) indicates that the point is in the direction opposite the angle \( \theta \) from the pole. The flexibility of polar coordinates allows for describing spirals, circles, and other curved forms more naturally than rectangular coordinates.

Coordinate Transformation

Coordinate transformation entails shifting from one coordinate system to another, in this case, converting from rectangular to polar coordinates or vice versa. Understanding this process is crucial for solving geometric problems in different contexts.

The transformation from rectangular coordinates \( (x, y) \) to polar coordinates involves trigonometric functions and the Pythagorean theorem. Similarly, to go from polar \( (r, \theta) \) back to rectangular, those same relations are employed but in reverse. When \( r \) is negative, a special consideration is taken, which rotates the point 180 degrees about the origin in the rectangular system, thus keeping the angle \( \theta \) the same but flipping the direction in which \( r \) is measured. This maneuver adheres to the symmetry of the coordinate systems and ensures the integrity of transformation across all quadrants and for all values of \( r \) and \( \theta \) .

Trigonometric Functions

Trigonometric functions are the tools that enable the conversion between rectangular and polar coordinates. They relate the angles of a triangle to the lengths of the triangle’s sides. The main trigonometric functions used in this context are sine (sin), cosine (cos), and tangent (tan).

In a right triangle, cosine relates the adjacent side to the hypotenuse, sine relates the opposite side to the hypotenuse, and tangent relates the opposite side to the adjacent side. When converting \( (x, y) \) to \( (r, \theta) \) or vice versa, we employ these functions to determine the necessary distances and angles.

For instance, to find the angle \( \theta \) using rectangular coordinates, one would use the arctangent function, defined as \( \arctan{\frac{y}{x}} \). And conversely, if we have the polar coordinates and wish to find \( x \) and \( y \) , cosine and sine are used respectively: \( x = r \cos{\theta} \) and \( y = r \sin{\theta} \). These functions are vital in mapping one system onto the other and form the backbone of coordinate conversion.