In Exercises 33–40, polar coordinates of a point are given. Find the rectangular coordinates of each point. (−4, π/2)

Verified step by step guidance

1. Recall the relationship between polar coordinates \((r, \theta)\) and rectangular coordinates \((x, y)\): \(x = r \cos{\theta}\) and \(y = r \sin{\theta}\).
2. Identify the given polar coordinates: \(r = -4\) and \(\theta = \frac{\pi}{2}\).
3. Calculate the \(x\)-coordinate using the formula: \(x = -4 \times \cos{\left(\frac{\pi}{2}\right)}\).
4. Calculate the \(y\)-coordinate using the formula: \(y = -4 \times \sin{\left(\frac{\pi}{2}\right)}\).
5. Combine the results 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 point (r, θ) specifies how far and in which direction the point lies relative to the origin.

Rectangular Coordinates

Rectangular coordinates (x, y) describe a point's position using horizontal and vertical distances from the origin along the x- and y-axes. Converting from polar to rectangular coordinates involves finding these x and y values.

Conversion Formulas between Polar and 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. These formulas use trigonometric functions to project the point onto the x- and y-axes.