Problem 89 - Precalculus with Limits

Question

In Exercises 85-108, convert the polar equation to rectangular form. \(\theta=2\pi/3\)

Step-by-Step Solution

Answer

The rectangular form equivalent for the polar equation \(\theta = \frac{2\pi}{3}\) is a line departing from the origin and making an angle of \(\frac{2\pi}{3}\) with the positive x-axis.

Step 1: Conversion Formulas

In polar coordinates, a point in the plane is represented by (r, theta). However, the same point in rectangular coordinates is represented by (x, y). To convert between these two systems, the following formulas are used: x = r cos(θ) and y = r sin(θ).

Step 2: Apply the Conversion

In the given equation, there is no 'r' value explicitly mentioned. This means that 'r' can take on any value. The angle \(\theta\) is fixed and equals \(\frac{2\pi}{3}\). However, this does not provide a direct x, y equivalent in the rectangular system, as we still need an 'r' value to calculate x and y.

Step 3: Rectangular Form

But if we still want to represent this in rectangular coordinate system we can use the simple representation when \(\theta = \frac{2\pi}{3}\), which is basically a line departing from the origin and making an angle of \(\frac{2\pi}{3}\) with the positive x-axis. This statement is the rectangular form equivalent.

Key Concepts

Polar Coordinates

When we talk about polar coordinates, we're looking at a two-dimensional coordinate system where each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point is commonly called the pole, similar to the origin in the Cartesian system, and the reference direction usually points to the right, aligning with the positive x-axis.

In polar coordinates, each point is represented by the pair \( (r, \theta) \), where \( r \) is the radial distance from the pole (equivalent to the origin in rectangular coordinates), and \( \theta \) is the angle measured in radians from the positive x-axis, going counterclockwise. A unique feature of polar coordinates is that they can represent points at infinity, where \( r = \infty \), which is not possible in rectangular coordinates.

The beauty of the polar system is its simplicity in dealing with curves and shapes that are inherently circular or radial, such as spirals or concentric circles, which can be more complex in rectangular coordinates.

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, are perhaps the most familiar coordinate system, where points are defined by their horizontal and vertical distances from a reference point. This system uses two axes, x and y, that intersect at a right angle at the point of origin. Each point in the plane is associated with a pair \( (x, y) \), where \( x \) is the distance along the horizontal axis, and \( y \) is the distance along the vertical axis.

Converting between polar and rectangular coordinates requires the use of trigonometry, since these two systems align at the origin but use different methods to locate points in space. Rectangular coordinates are especially useful in scenarios where linear distances and orthogonal relations are essential, like in geometry, architecture, and many engineering fields.

Trigonometric Conversion

To bridge the gap between polar and rectangular coordinates, we need trigonometric conversion, which is fundamentally using the properties of right triangles to relate polar and Cartesian systems. This is because any point in polar coordinates can form a right triangle with the x-axis and a vertical line dropping from the point to the x-axis.

The conversion is rooted in the circular functions sine and cosine. The formulas \( x = r \cos(\theta) \) and \( y = r \sin(\theta) \) are the keystones of this conversion. They allow us to express the radial distance and angle of polar coordinates in terms of the horizontal and vertical distances of rectangular coordinates. In cases where the polar equation represents a line, as in our exercise with \( \theta = \frac{2\pi}{3} \), the conversion describes a line in the rectangular system, signifying all points that make the angle \( \frac{2\pi}{3} \) with the positive x-axis.