For each pair of polar coordinates, (c) give the rectangular coordinates for the point. See Examples 1 and 2(a).


(4 , 3π/2)

Verified step by step guidance

1. Step 1: Understand the relationship between polar and rectangular coordinates. Polar coordinates are given as (r, θ), where r is the radius (distance from the origin) and θ is the angle in radians from the positive x-axis. Rectangular coordinates are given as (x, y).
2. Step 2: Use the conversion formulas to transform polar coordinates to rectangular coordinates. The formulas are: x = r * cos(θ) and y = r * sin(θ).
3. Step 3: Substitute the given polar coordinates into the conversion formulas. Here, r = 4 and θ = \(\frac{3\pi}{2}\).
4. Step 4: Calculate x using the formula x = r * cos(θ). Substitute r = 4 and θ = \(\frac{3\pi}{2}\) into the equation.
5. Step 5: Calculate y using the formula y = r * sin(θ). Substitute r = 4 and θ = \(\frac{3\pi}{2}\) into the equation.

Key Concepts

Polar Coordinates

Polar coordinates represent a point in a two-dimensional space using a distance from a reference point (the origin) and an angle from a reference direction (usually the positive x-axis). The format is (r, θ), where 'r' is the radial distance and 'θ' is the angle in radians. Understanding this system is crucial for converting to rectangular coordinates.

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, express a point in a two-dimensional space using two perpendicular axes, typically labeled x and y. The coordinates are given in the form (x, y). Converting from polar to rectangular coordinates involves using the relationships x = r * cos(θ) and y = r * sin(θ).

Conversion Formulas

To convert polar coordinates to rectangular coordinates, specific trigonometric formulas are applied. For a point given in polar form (r, θ), the conversion is done using x = r * cos(θ) and y = r * sin(θ). These formulas utilize the angle and the radius to find the corresponding x and y values in the Cartesian plane.