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


(3 , 120°)

Verified step by step guidance

1. Recall the relationship between polar coordinates \((r, \theta)\) and rectangular coordinates \((x, y)\), where \(x = r \cos(\theta)\) and \(y = r \sin(\theta)\).
2. Identify the given polar coordinates: \(r = 3\) and \(\theta = 120^\circ\).
3. Convert the angle \(\theta\) from degrees to radians if necessary, or use the degree mode directly in trigonometric functions. Here, we can use \(120^\circ\) directly.
4. Calculate the \(x\)-coordinate using the formula \(x = 3 \times \cos(120^\circ)\).
5. Calculate the \(y\)-coordinate using the formula \(y = 3 \times \sin(120^\circ)\).

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 typically 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 maps the point from a radial system to the Cartesian plane, facilitating easier computation and visualization.

Trigonometric Functions and Angle Measurement

Understanding sine and cosine functions and how to evaluate them at specific angles (like 120°) is essential. Angles in degrees must be converted to radians if necessary, and knowledge of the unit circle helps determine the sign and value of these functions.