Problem 35 - 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. \(\left(-4.1, -0.5\right)\)

Step-by-Step Solution

Answer

The rectangular coordinates obtained from the polar coordinates (-4.1, -0.5) will be the calculated x and y values, both rounded to two decimal places.

Step 1: Understand Polar Coordinates

Polar coordinates (r,θ) are represented as a radius and an angle. Here, r = -4.1 and θ = -0.5 are given.

Step 2: Convert Polar to Rectangular Coordinates

The formulas to convert polar coordinates (r,θ) to rectangular coordinates (x,y) are \(x = r * cos(θ)\) and \(y = r * sin(θ)\). Applying these formulas, calculate x and y.

Step 3: Calculate x

Using the formula \(x = r * cos(θ)\), plug in the provided values: \(x = -4.1 * cos(-0.5)\). Use a graphing utility to compute the result.

Step 4: Calculate y

Next, calculate y value using the formula \(y = r * sin(θ)\), thus \(y = -4.1 * sin(-0.5)\). Compute the result with a graphing utility.

Step 5: Round the Results

Now, round the computed x and y values to two decimal places as directed in the exercise.

Key Concepts

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, represent points in a plane using two values: x and y. We often visualize this as a spot on a grid where each point has a specific horizontal distance (x) and a vertical distance (y) from the origin. In essence, these coordinates form a way to pinpoint any location on a 2D plane.
The x-coordinate denotes the horizontal direction from the origin—the center point of the grid—while the y-coordinate denotes the vertical movement. For example, the point (3, 4) sits 3 units right and 4 units up from the origin.

• This system is foundational in various fields like physics and engineering, where precise measurement and plotting are crucial.

Understanding rectangular coordinates is essential since many real-world applications utilize these values to describe locations and guide navigation.

Coordinate Conversion

Coordinate conversion involves transforming points from one system to another. In this exercise, we are converting from polar to rectangular coordinates, commonly encountered in mathematics. Polar coordinates, given as (r, θ), use a radius and an angle to describe a point relative to the origin. This is particularly useful in scenarios involving circular or rotational symmetry.
Converting these to rectangular coordinates allows for simpler integration with other mathematical models. The formulas used for conversion are:

• The x-coordinate: \(x = r \cdot \cos(θ)\)
• The y-coordinate: \(y = r \cdot \sin(θ)\).

By calculating these, you transform the radial and angular representation into a simpler, more universally applicable form. This process enables easier operations such as addition, subtraction, or solving for intersections of lines and curves.

Trigonometric Functions

Trigonometric functions, namely sine and cosine, are vital tools in converting polar coordinates to rectangular form. These functions relate angles of a right triangle to the lengths of its sides—a concept extending to rotations and circles in a coordinate plane.
The cosine function, represented as \(\cos(θ)\), gives the horizontal component of a point, effectively measuring how far left or right from the origin the point lies. Similarly, the sine function, \(\sin(θ)\), gives the vertical component, showing how far up or down the point is.
Using these functions allows us to break down a polar vector, or line of fixed radius and angle, into x and y segments in the rectangular system.

• This breakdown is crucial in numerous applications such as calculating wave motions, engineering designs, and even simple physics problems involving projectile motion.
• It also helps in simplifying complex problems into more manageable, bite-sized calculations by leveraging the power of trigonometry.

Mastery of trigonometric functions not only aids in coordinate conversion tasks but also enhances overall mathematical fluency.