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


(5 , ―60°)

Verified step by step guidance

1. Recall that to convert from polar coordinates \((r, \theta)\) to rectangular coordinates \((x, y)\), we use the formulas: \(x = r \cos(\theta)\) \(y = r \sin(\theta)\).
2. Identify the given polar coordinates: here, \(r = 5\) and \(\theta = -60^\circ\).
3. Substitute the values into the formulas: \(x = 5 \cos(-60^\circ)\) \(y = 5 \sin(-60^\circ)\).
4. Use the fact that cosine is an even function and sine is an odd function, so \(\cos(-60^\circ) = \cos(60^\circ)\) \(\sin(-60^\circ) = -\sin(60^\circ)\).
5. Evaluate the trigonometric values \(\cos(60^\circ)\) and \(\sin(60^\circ)\) using known exact values or a unit circle, then multiply by 5 to find \(x\) and \(y\).

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 is expressed as (r, θ), 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 translates the point from a radius and angle to Cartesian coordinates on the xy-plane.

Trigonometric Functions and Angle Measurement

Understanding sine and cosine functions and how they relate to angles is essential. Angles can be negative or positive, and the trigonometric values determine the sign and position of the point in the rectangular coordinate system.