91Ó°ÊÓ

First, let's analyze our given equation: \(r^{2}=\sin 2 \theta\). Since the equation describes a polar curve, we can think about how the radius (r) changes as the angle (θ) changes. Notice that the range of the sine function is between -1 and 1, i.e., -1≤sin(2θ)≤1. Therefore, the range of the radius squared is also between 0 and 1, i.e., 0≤\(r^{2}\)≤1. Now, let's examine how the radius changes with the angle in more detail.

To find the key points, we'll study the behavior of the curve for different angle values (θ). 1. When θ = 0, sin(2θ) = sin(0) = 0, thus \(r^{2}\) = 0, and r = 0. 2. When θ = \(π/4\), sin(2θ) = sin(π/2) = 1, thus \(r^{2}\) = 1, and r = ±1. 3. When θ = \(π/2\), sin(2θ) = sin(π) = 0, thus \(r^{2}\) = 0, and r = 0. 4. When θ = \(3π/4\), sin(2θ) = sin(3π/2) = -1, however, the radius squared cannot be negative, so there is no solution for this angle. 5. When θ = π, sin(2θ) = sin(2π) = 0, thus \(r^{2}\) = 0, and r = 0. Now we have the key points for some important θ-values. Let's plot these points and sketch the curve.

Using the key points found in Step 2: 1. (θ = 0, r = 0) Start at the origin. 2. (θ = \(π/4\), r = ±1) This tells us that at angle π/4, we have two points on the curve with radius 1. These points are symmetric with respect to the origin. 3. (θ = \(π/2\), r = 0) The curve returns to the origin at angle π/2. 4. (θ = \(3π/4\)) Our curve doesn't have a solution for this angle. 5. (θ = π, r = 0) The curve returns to the origin at angle π. As θ increases from 0 to \(π/4\), the curve goes from the origin to a maximum amplitude of 1 in both the positive and negative radial directions. Then, the curve returns to the origin at θ = \(π/2\). The curve is symmetric with respect to the origin. Continue sketching in the same way for all other values of θ. Additionally, since the sine function has a period of \(2 \pi\), the polar curve will also repeat itself every \(2 \pi\). So, the whole curve will be described between 0 and \(2 \pi\), and the shape will keep repeating for greater angle values.