EXAMPLE 3 Identifying Graphs Graph the sets of points whose polar coordinates satisfy the following conditions. (a) 1leqslant rleqslant 2 and

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EXAMPLE 3 Identifying Graphs Graph the sets of points whose polar coordinates satisfy the following conditions. (a) 1leqslant rleqslant 2 and

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To graph the set of points in polar coordinates that satisfy the condition \(1 \leqslant r \leqslant 2\), we need to understand what this inequality represents in terms of the polar coordinate system.

In polar coordinates, \(r\) represents the radius or distance from the origin, and \(\theta\) represents the angle measured counterclockwise from the positive x-axis.

The inequality \(1 \leqslant r \leqslant 2\) means that we are looking for all points where the radius \(r\) is between 1 and 2, inclusive. This can be interpreted as follows:

- The point at \(r = 1\) is a circle centered at the origin with radius 1.
- The point at \(r = 2\) is a circle centered at the origin with radius 2.

Therefore, the set of points satisfying \(1 \leqslant r \leqslant 2\) will be the annular region (ring-shaped area) between these two circles, including the boundaries.

To graph this:

1. Draw a circle with radius 1 centered at the origin.
2. Draw another circle with radius 2 centered at the origin.
3. Shade the region between these two circles, including the circles themselves.

This shaded region represents all the points whose polar coordinates satisfy \(1 \leqslant r \leqslant 2\).

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EXAMPLE 3 Identifying Graphs Graph the sets of points whose polar coordinates satisfy the following conditions. (a) 1leqslant rleqslant 2 and

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