Problem 20 - Precalculus with Limits

Question

In Exercises 19-28, a point in polar coordinates is given. Convert the point to rectangular coordinates. \(\left(3, 3\pi/2\right)\)

Step-by-Step Solution

Answer

The rectangular coordinates for the given polar coordinates \((3, 3π/2)\) are (0, -3).

Step 1: Identify the given polar coordinates

The point is given in polar coordinates as (r, θ) = (3, 3π/2). Here, r is the distance from the origin, which is 3, and θ is the angle measured in radians, which is 3π/2.

Step 2: Apply the conversion formulas

Apply the formulas \(x=r*cos(θ)\) and \(y=r*sin(θ)\) to get the rectangular coordinates. Substitute r = 3 and θ = 3π/2 into these equations to get the rectangular coordinates.

Step 3: Calculation

On calculation, \(x=3*cos(3π/2) = 3*0 = 0\) and \(y=3*sin(3π/2) = 3*-1 = -3\) . Therefore, the rectangular coordinates for the given polar coordinates are (0, -3).

Key Concepts

Polar Coordinates

Polar coordinates are a way of expressing points in the plane using a distance from a reference point and an angle from a reference direction. The reference point is usually called the origin, and the reference direction is typically the positive x-axis. A point in polar coordinates is represented as \((r, \theta)\), where:

• \(r\) is the radial distance from the origin.
• \(\theta\) is the angle from the positive x-axis, measured in radians or degrees.

These coordinates are particularly useful in situations where the focus is on direction and distance from a central point, such as in navigation or circular motion. For instance, in our exercise, the point \((3, \frac{3\pi}{2})\) indicates a distance of 3 units from the origin, at an angle of \(\frac{3\pi}{2}\) radians, which is equivalent to 270 degrees.

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, describe a point using two perpendicular axes: the x-axis and the y-axis. Any point in the plane can be described as \((x, y)\) where:

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

These coordinates are incredibly versatile—particularly useful in algebraic contexts, drawing graphs, and solving geometric problems. For the polar point \((3, \frac{3\pi}{2})\), after converting, we find the rectangular coordinates are \((0, -3)\). This result shows the point is located on the negative y-axis, 3 units away from the origin. Understanding how to switch between polar and rectangular coordinates allows for effective use of the most convenient system for any given problem.

Trigonometric Functions

Trigonometric functions play a crucial role in the conversion between polar and rectangular coordinates. The functions \(\cos(\theta)\) and \(\sin(\theta)\) relate the angle to a point's position on a circle's circumference based on its radius.To convert from polar to rectangular coordinates, we use these formulas:

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

For example, using \(r = 3\) and \(\theta = \frac{3\pi}{2}\):

• \(x = 3 \times \cos(\frac{3\pi}{2}) = 0\)
• \(y = 3 \times \sin(\frac{3\pi}{2}) = -3\)

These calculations correctly yield the rectangular coordinates \((0, -3)\). Understanding trigonometric functions like sine and cosine is fundamental, as they help bridge the gap between circular and linear systems, enabling seamless transitions between different methods of describing space.