Problem 28 - Precalculus with Limits

Question

In Exercises 19-28, a point in polar coordinates is given. Convert the point to rectangular coordinates. \(\left(-2, 5.76\right)\)

Step-by-Step Solution

Answer

The rectangular (Cartesian) coordinates of the given point in polar coordinates is approximately (-1.68, -1.79).

Step 1: Identify the Polar Coordinates

Given polar coordinates are (-2, 5.76) where -2 is the distance from the origin (R) and 5.76 is the angle (θ) in radians.

Step 2: Convert the Polar Coordinates to Rectangular Coordinates

Apply the formulas \(x = R * \cos(\theta)\) and \(y = R * \sin(\theta)\). Since R is negative, it means that the point lies in the opposite direction of the angle θ. Adjust the θ by adding or subtracting pi (π) depending on its current position: θ' = θ + π = 5.76 + 3.14 = 8.9. Now apply the formulas: x = -2 * \cos(8.9) and y = -2 * \sin(8.9). Calculate the values.

Step 3: Final Rectangular Coordinates

After computing the values using a calculator, the rectangular coordinates are approximately (\(x, y) = (-1.68, -1.79)\).

Key Concepts

Polar Coordinates

Polar coordinates represent a point in a plane using a distance from a reference point and an angle from a reference direction. Typically,- The reference point is known as the pole (often the origin in the coordinate system).- The distance from the pole to the point is called the radial coordinate (R), and it can be positive or negative.- The angle, usually measured from the positive x-axis, is called the angular coordinate (θ). The angle can be given in radians or degrees.

In polar coordinates, a point is represented as (R, θ). For example, the point \((-2, 5.76)\) has a radial distance of -2, meaning it lies opposite to the given angle direction, and the angle of 5.76 radians. To locate a point using polar coordinates, you move R units toward the angle θ. If R is negative, move in the opposite direction to angle θ. This unique aspect of polar coordinates allows flexibility in plotting points on a plane, especially useful in scenarios involving curves and circular paths.

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, are the conventionally used system where each point in a plane is determined by a pair of numbers (x, y). Each number in the pair corresponds to a distance along one of the two perpendicular axes. In this system:- The horizontal axis is usually labeled as the x-axis.- The vertical axis is usually labeled as the y-axis.

In rectangular coordinates, a point is described by the distance from the origin along these two axes. The transition from polar to rectangular coordinates involves using trigonometric functions, specifically using the formulas:- \(x = R \cdot \cos(θ)\)- \(y = R \cdot \sin(θ)\)

These formulas are derived from the trigonometric principles in right triangles. When converting, if the radial coordinate is negative as in the example given, it indicates moving in the opposite direction from the angle θ. Understanding the conversion between these two systems helps in visualizing and solving problems in different formats, giving a versatile tool in mathematics.

Trigonometric Functions

Trigonometric functions are essential in converting polar coordinates to rectangular coordinates. These functions, \(\cos\) and \(\sin\), relate angles of a triangle to the ratios of its sides in right-angled triangles. In the context of coordinate conversion:- \(\cos(θ)\) provides the ratio for the adjacent side over the hypotenuse.- \(\sin(θ)\) gives the ratio for the opposite side over the hypotenuse.

When you have a point given in polar coordinates (R, θ), use these functions to find the x and y in rectangular form by utilizing:- \(x = R \cdot \cos(θ)\) derives from projecting the point onto the x-axis.- \(y = R \cdot \sin(θ)\) comes from projecting the point onto the y-axis.

These trigonometric calculations ensure that you accurately translate the radial and angular measurements of polar coordinates into the x and y axes, allowing you to work seamlessly between different coordinate systems in geometry and physics.