Problem 41 - Precalculus

Question

Use a graphing utility to find the rectangular coordinates of the point given in polar coordinates. Round your results to two decimal places. $$(-4.1,-0.5)$$

Step-by-Step Solution

Answer

The rectangular coordinates of the point given, when rounded to two decimal places, are approximately (3.83, 2.00).

Step 1: Recognize the polar coordinates

In this task, we are given the polar coordinates as \((-4.1, -0.5)\). Notice that the first value represents the radial distance, r, from the origin and the second value is the angle, \( \theta \), in radian measure.

Step 2: Apply the conversion formula

To convert polar coordinates to rectangular ones, we use the formulas \(x = r \cos(\theta)\) and \(y = r \sin(\theta)\). Here, r = -4.1 and \( \theta = -0.5 \) radian. Thus, substituting the values in the given formulas, we find that \(x = -4.1 \cos(-0.5)\) and \(y = -4.1 \sin(-0.5)\)

Step 3: Calculate the x and y values

Using a graphing utility or a calculator, we find that \(x = -4.1 \cos(-0.5) \approx 3.83\) and \(y = -4.1 \sin(-0.5) \approx 2.00\). Remember that we round the results to two decimal places, as the problem statement says. Hence, the rectangular coordinates are approximately (3.83, 2.00).

Key Concepts

Polar Coordinates

Polar coordinates are a way of representing points in the plane, using a combination of distance and angle. This system is particularly useful for circular and rotational movements.
In polar coordinates, each point is determined by:

• The radial distance \(r\), which is the distance from the point to the origin.
• The angle \(\theta\), measured in radians from the positive x-axis.

Polar coordinates are often written in the form \((r, \theta)\).
It’s important to remember that angles in polar coordinates can have multiple representations due to their cyclical nature. For example, an angle of \(-0.5\) radians means that it is measured in the clockwise direction.

Rectangular Coordinates

Rectangular coordinates, often called Cartesian coordinates, are a more familiar system to many. They define points by their horizontal and vertical distances from the origin, forming a rectangular grid.
Each point in rectangular coordinates is expressed as \((x, y)\):

• \(x\) is the horizontal distance from the origin.
• \(y\) is the vertical distance from the origin.

The transformation from polar to rectangular coordinates involves trigonometric functions, leveraging both cosine and sine to calculate the \(x\) and \(y\) coordinates respectively.
This system is widely used because it maps naturally to our perception of space with its straightforward grid-like layout.

Trigonometric Functions

Trigonometric functions are mathematical functions that relate the angles of a triangle to the lengths of its sides. They are crucial in converting polar coordinates to rectangular coordinates.
The primary trigonometric functions used in this context are:

• Cosine \(\cos(\theta)\), which gives the horizontal component (or \(x\)-coordinate).
• Sine \(\sin(\theta)\), which gives the vertical component (or \(y\)-coordinate).

To convert, you use these functions with the following formulas:

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

These functions allow you to project the radial distance onto the x and y axes, effectively transforming a problem in circular coordinates into a linear one which is often easier to interpret visually.