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. Recall the relationship between polar coordinates \((r, \theta)\) and rectangular coordinates \((x, y)\), which is given by the formulas: \(x = r \cos(\theta)\) and \(y = r \sin(\theta)\).
2. Identify the given polar coordinates: \(r = 4\) and \(\theta = \frac{3\pi}{2}\).
3. Calculate the \(x\)-coordinate using the formula: \(x = 4 \times \cos\left(\frac{3\pi}{2}\right)\).
4. Calculate the \(y\)-coordinate using the formula: \(y = 4 \times \sin\left(\frac{3\pi}{2}\right)\).
5. Combine the results from steps 3 and 4 to write the rectangular coordinates as \((x, y)\).

Key Concepts

Polar Coordinates

Polar coordinates represent a point in the plane using a distance from the origin (radius r) and an angle θ measured from the positive x-axis. The pair (r, θ) specifies the location uniquely, where r ≥ 0 and θ is usually in radians.

Conversion from Polar to Rectangular Coordinates

To convert polar coordinates (r, θ) to rectangular coordinates (x, y), use the formulas x = r cos(θ) and y = r sin(θ). This translates the point from a radius-angle format to Cartesian coordinates on the xy-plane.

Trigonometric Functions and Angle Measurement

Understanding sine and cosine values for standard angles, especially multiples of π/2, is essential. For example, cos(3π/2) = 0 and sin(3π/2) = -1, which helps in accurately computing the rectangular coordinates.