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


(―3 , ―210°)

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: \(r = -3\) and \(\theta = -210^\circ\).
3. Convert the angle \(\theta\) to a standard position if needed. Since \(-210^\circ\) is negative, you can add \(360^\circ\) to find a positive coterminal angle: \(-210^\circ + 360^\circ = 150^\circ\).
4. Calculate the rectangular coordinates using the formulas: \(x = -3 \cos(150^\circ)\) and \(y = -3 \sin(150^\circ)\).
5. Evaluate the cosine and sine values for \(150^\circ\) and multiply by \(-3\) to find the exact rectangular coordinates \((x, y)\).

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 can be positive or negative, and θ is usually given in degrees or 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 transformation translates the point from a radius-angle format to Cartesian coordinates on the xy-plane.

Handling Negative Radius and Angle Measures

A negative radius means the point lies in the direction opposite to the angle θ, effectively adding 180° to θ. Negative angles indicate rotation clockwise from the positive x-axis. Properly adjusting these values ensures accurate conversion to rectangular coordinates.