Problem 31 - Precalculus with Limits

Question

In Exercises 29-36, use a graphing utility to find the rectangular coordinates of the point given in polar coordinates. Round your results to two decimal places. \(\left(-4.5, 1.3\right)\)

Step-by-Step Solution

Answer

The rectangular coordinates of the point with polar coordinates \((-4.5, 1.3)\) are calculated using the formulas \(x = r \cdot \cos(\theta)\) and \(y = r \cdot \sin(\theta)\). The exact values for \(x\) and \(y\) will depend on the use of a calculator.

Step 1: Identify the given polar coordinates

The polar coordinates provided in the problem are \(-4.5\) for \(r\) and \(1.3\) for \(\theta\). Note that \(r\) is negative, which means that the point is located in the opposite direction.

Step 2: Apply the conversion formulas

Next, plug the polar coordinates into the conversion formulas to get the rectangular coordinates. We have \(x = r \cdot \cos(\theta)\) and \(y = r \cdot \sin(\theta)\). Substituting the given values, \(x = -4.5 \cdot \cos(1.3)\) and \(y = -4.5 \cdot \sin(1.3)\).

Step 3: Calculate the rectangular coordinates

Using a calculator, compute the values of \(x\) and \(y\) rounding to two decimal places. These are the rectangular coordinates of the given polar point.

Key Concepts

Polar Coordinates

Polar coordinates are an essential concept in mathematics, allowing us to describe points on a plane using a distance and an angle. In the polar coordinate system, a point is represented as \((r, \theta)\). Here,

• \(r\) denotes the radial distance from the origin (or center), and
• \(\theta\) is the angular displacement measured in radians or degrees from the positive x-axis.

The polar coordinate system is particularly useful in scenarios where circular or rotational patterns are involved. Unlike the more straightforward rectangular coordinates, which are based on fixed units along the x and y axes, polar coordinates can seem counterintuitive since they involve angles and radii.
It's important to note that a negative \(r\) value implies that the direction is reversed from the origin along the angle \(\theta\). This reversal can place the point in a different quadrant from what one might initially assume.

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, utilize a grid system with two perpendicular axes, labeled as the x-axis (horizontal) and y-axis (vertical). A point's location in this system is defined by its distances \( (x, y) \) from the two axes.

• The x-coordinate measures the horizontal distance from the y-axis.
• The y-coordinate indicates the vertical distance from the x-axis.

Converting polar coordinates to rectangular coordinates is often necessary for graphing utilities and some calculations. This conversion is achieved using trigonometric functions as part of a well-established formula:

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

These equations transform the radially and angularly defined polar coordinates into the linear dimensions required by the rectangular system. Recognizing how each coordinate system offers unique advantages for different types of problems highlights the value of both approaches in mathematical analysis.

Trigonometric Functions

Trigonometric functions are the backbone of converting between polar and rectangular coordinates. These functions connect angles and ratios in right triangles, and they are pivotal in various areas of mathematics. Three essential trigonometric functions are sine, cosine, and tangent:

• The cosine function \((\cos)\) relates the adjacent side of a right triangle to its hypotenuse.
• The sine function \((\sin)\) compares the opposite side of the triangle to its hypotenuse.
• Tangent \((\tan)\) involves the ratio of the opposite to the adjacent sides.

In the conversion process from polar to rectangular coordinates, cosine and sine are crucial. When you know the radial distance \(r\) and the angle \(\theta\), you multiply \(r\) by \(\cos(\theta)\) and \(\sin(\theta)\) to find the x and y coordinates, respectively. This transformation is what takes us from a world of circles and angles into one of straight lines and scalar distances. Understanding these functions helps demystify the conversion between these two coordinate systems, facilitating clearer and more efficient problem-solving.