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


(1 , 45°)

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: \(r = 1\) and \(\theta = 45^\circ\).
3. Convert the angle \(\theta\) from degrees to radians if necessary, but since \(45^\circ\) is a common angle, you can use the exact trigonometric values directly.
4. Calculate the \(x\)-coordinate using \(x = 1 \times \cos(45^\circ)\).
5. Calculate the \(y\)-coordinate using \(y = 1 \times \sin(45^\circ)\).

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 is essential for conversion, as they relate angles to ratios of sides in right triangles. Angles must be correctly interpreted in degrees or radians to apply these functions accurately during coordinate conversion.