Problem 67 - Precalculus : Building Concepts and Connections

Question

A patented device converts a radar signal given in polar coordinates to a format in rectangular coordinates so that it is better suited to display in a televisiontype display device. If a radar signal is at the point \(\left(3,-\frac{2 \pi}{3}\right)\) find the exact values of the corresponding rectangular coordinates in the television display.

Step-by-Step Solution

Answer

The exact values of the corresponding rectangular coordinates in the television display are (-1.5, -2.598).

Step 1: Identify the given polar coordinates

The given polar coordinates are \(r = 3\) and \(\theta = -\frac{2 \pi}{3}\).

Step 2: Apply the conversion formulas

Applying the conversion formulas, compute \(x = r \cos(\theta)\) and \(y = r \sin(\theta)\). This leads to \(x = 3 \cos\left(-\frac{2 \pi}{3}\right)\) and \(y = 3 \sin\left(-\frac{2 \pi}{3}\right)\).

Step 3: Compute the coordinates

Using a calculator or trigonometric tables for precise values, we find \(x = 3 \cos\left(-\frac{2 \pi}{3}\right) = -1.5\) and \(y = 3 \sin\left(-\frac{2 \pi}{3}\right) = -2.598\).

Key Concepts

Polar Coordinates

Polar coordinates are an alternative to the more familiar rectangular coordinate system for representing positions on a two-dimensional plane. While rectangular coordinates use horizontal and vertical components, known as x and y, to describe the location of a point, polar coordinates use a radius and an angle to express this location.

In polar coordinates, a point is represented by a pair \( (r, \theta) \) where \( r \) is the distance from the origin (the pole) to the point and \( \theta \) is the angle between the positive x-axis and the line connecting the origin to the point. This system is especially useful in situations where phenomena naturally form around a central point, such as the propagation of radar signals or the orbits of celestial bodies.

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, are the most commonly used coordinate system. They divide the plane into four quadrants using two perpendicular axes: the x-axis horizontally and the y-axis vertically. Every point in the plane is identified by an ordered pair \( (x, y) \) where \( x \) is the distance to the right (or left) of the y-axis, and \( y \) is the distance above (or below) the x-axis.

Rectangular coordinates are widely favored for their simplicity, especially in conventional display formats such as computer and television screens. They are intuitive for constructing geometric shapes and for performing algebraic computations.

Conversion Formulas

Conversion formulas bridge the gap between polar and rectangular coordinates, allowing for the translation of points from one system to the other. The formulas derive from the definitions of sine and cosine in right-angled triangles and trigonometric identities.

The conversion from polar to rectangular coordinates is given by the formulas:

• \(x = r \cos(\theta)\)
• \(y = r \sin(\theta)\)

Conversely, converting from rectangular to polar coordinates involves these formulas:

• \(r = \sqrt{x^2 + y^2}\)
• \(\theta = \arctan\left(\frac{y}{x}\right)\)

Understanding these relationships is crucial for fields that utilize both systems, such as engineering and physics.

Trigonometric Functions

Trigonometric functions are fundamental to the relationship between the sides and angles of triangles, and they are vital for converting between polar and rectangular coordinates. The primary trigonometric functions are sine (sin), cosine (cos), and tangent (tan).

The functions are defined as ratios of sides in a right-angled triangle: sine is opposite over hypotenuse, cosine is adjacent over hypotenuse, and tangent is opposite over adjacent. These functions also describe the relationships of the angles to the unit circle, where the hypotenuse equals one. In the context of conversion, knowing the exact values of trigonometric functions for common angles, such as \(-\frac{2 \pi}{3}\), is essential for accurate calculation of coordinates.