To sketch the curve for $r = 2\sin(x)$ in polar coordinates, let's go through the key points.

Step-by-Step Guide:

1. Understanding the Polar Equation:

   • In polar coordinates, the equation $r = 2\sin(x)$ represents a relationship between the radial distance $r$ from the origin and the angle $x$, which is the polar angle (in radians).
   • For each value of $x$, $r$ gives the distance from the origin (pole).

2. Key Features:

   • The curve repeats every $2\pi$, since $\sin(x)$ has a period of $2\pi$.
   • The values of $r$ vary depending on the angle $x$. Specifically:
     • When $x = 0$, $r = 0$.
     • When $x = \frac{\pi}{2}$, $r = 2$.
     • When $x = \pi$, $r = 0$.
     • When $x = \frac{3\pi}{2}$, $r = -2$ (meaning the point is plotted on the opposite side of the origin).
   • Since $r$ can be negative, the curve reflects across the origin for certain angles.

3. Shape of the Curve:

   • The curve is symmetric about the polar axis (horizontal axis), and the resulting shape is known as a limacon.
   • It has an inner loop due to the negative values of $r$.

4. Plotting Important Points:

   • At $x = 0$, $r = 0$.
   • At $x = \frac{\pi}{2}$, $r = 2$, so plot a point 2 units away from the origin.
   • At $x = \pi$, $r = 0$.
   • At $x = \frac{3\pi}{2}$, $r = -2$, meaning the point is 2 units away but on the opposite side of the origin.

5. Final Shape:

   • The curve forms a limacon with no dimple and a loop. It starts at the origin, loops outward to a maximum of 2 units, and then reflects to form a symmetrical shape.

Would you like me to generate a sketch of this curve for you?