Problem 4 - Precalculus with Limits

Question

The polar coordinates \((r, 0)\) are related to the rectangular coordinates \((x, y)\) as follows: \(x =\) ________ \(\quad y =\) ________ \(\quad \tan\ \theta =\) ________ \(\quad r^2 =\) ________

Step-by-Step Solution

Answer

The rectangular coordinates are: x = r, y = 0. The tan θ is 0 and \(r^{2} = r^{2}\).

Step 1: Find x

Since \(x=r\cos\theta\), and given that \(\theta=0\), cos(0)=1. Therefore, \(x=r*1=r\)

Step 2: Find y

Since \(y=r\sin\theta\), and given that \(\theta=0\), sin(0)=0. Therefore, \(y=r*0=0\)

Step 3: Find tan θ

Since \(\tan\theta= \frac{y}{x}\), and we know y=0 and x=r, then \(\tan\theta= \frac{0}{r} = 0\)

Step 4: Find r^2

Since \(r^{2}= x^{2} + y^{2}\), and we know y=0 and x=r, then \(r^{2}= r^{2}+ 0^{2} = r^{2}\)

Key Concepts

Rectangular Coordinates

In mathematics, rectangular coordinates, also known as Cartesian coordinates, are used to uniquely determine the location of a point in a plane using a pair of numerical values. These coordinates are expressed as \(x, y\), where \x\ is the horizontal distance of the point from the origin, and \y\ is the vertical distance. The origin, where both \x\ and \y\ equal zero, is the center point of the coordinate plane.

To determine these coordinates from polar coordinates, we use specific relationships:

• The formula for \x\ is given by \x = r\cos\theta\, where \r\ is the radial distance from the origin and \theta\ is the angle from the positive x-axis.
• For \y\, the formula is \y = r\sin\theta\.

When given \(r, \theta=0\), the conversion to rectangular coordinates simplifies since \cos(0) = 1\ and \sin(0) = 0\, thus \(x = r\) and \(y = 0\). These simple conversions reflect the straightforward relationship between these two coordinate systems.

Trigonometric Functions

Trigonometric functions relate the angles of a triangle to the lengths of its sides. They are fundamental in converting polar coordinates to rectangular coordinates.

In the context of coordinate conversion, we commonly use two trigonometric functions - cosine and sine - to express rectangular coordinates in terms of polar coordinates:

• \cos\theta\ measures the cosine of an angle \theta\, which is the ratio of the length of the adjacent side to the hypotenuse in a right-angled triangle. This is used in the formula \(x = r\cos\theta\).
• \sin\theta\ represents the sine of \theta\, which is the ratio of the length of the opposite side to the hypotenuse. This gives us the equation \(y = r\sin\theta\).

These functions are instrumental in translating angles into linear distances on the coordinate plane. When \theta = 0\ degrees, as seen in the problem, cosine equals 1 and sine equals 0, simplifying calculations significantly.

Coordinate Conversion

Coordinate conversion involves translating one type of coordinate system into another. Specifically, converting between polar and rectangular coordinates is a common task, especially in fields such as physics and engineering.

Polar coordinates \(r, \theta\) specify a point's position based on its distance from a fixed central point (the origin) and the angle from a fixed direction (the positive x-axis). The steps to convert these into rectangular coordinates \(x, y\) typically include:

• Using \(x = r\cos\theta\) to find the horizontal position.
• Applying \(y = r\sin\theta\) to determine the vertical position.

Additionally, you can determine the angle with \ \tan\theta = \frac{y}{x} \, and the magnitude \ r^2 = x^2 + y^2 \. This duality allows for flexible problem-solving approaches, depending on which system is more convenient for a given problem.