For a constant $c,$ in cylindrical coordinates $(r,\theta,z),$ find the shape described by the equation
\[\theta = c.\](A) Line
(B) Circle
(C) Plane
(D) Sphere
(E) Cylinder
(F) Cone

Enter the letter of the correct option.
In cylindrical coordinates, $\theta$ denotes the angle a point makes with the positive $x$-axis.  Thus, for a fixed angle $\theta = c,$ all the points lie on a plane.  The answer is $\boxed{\text{(C)}}.$  Note that we can obtain all points in this plane by taking $r$ negative.

[asy]
import three;
import solids;

size(200);
currentprojection = perspective(6,3,2);
currentlight = (1,0,1);
real theta = 150;

draw((0,0,0)--(-2,0,0));
draw((0,0,0)--(0,-2,0));
draw(surface((Cos(theta),Sin(theta),1)--(Cos(theta),Sin(theta),-1)--(Cos(theta + 180),Sin(theta + 180),-1)--(Cos(theta + 180),Sin(theta + 180),1)--cycle), gray(0.7),nolight);
draw((0,0,0)--(2,0,0));
draw((0,0,0)--(0,2,0));
draw((0,0,-1.5)--(0,0,1.5));
draw((1.5*Cos(theta),1.5*Sin(theta),0)--(1.5*Cos(theta + 180),1.5*Sin(theta + 180),0));
draw((0.5,0,0)..(0.5*Cos(theta/2),0.5*Sin(theta/2),0)..(0.5*Cos(theta),0.5*Sin(theta),0),red,Arrow3(6));
draw((0,0,0)--(0,-1,0),dashed);
draw((0,0,0)--(-2,0,0),dashed);

label("$\theta$", (0.7,0.6,0), white);
label("$x$", (2,0,0), SW);
label("$y$", (0,2,0), E);
label("$z$", (0,0,1.5), N);
label("$\theta = c$", (Cos(theta),Sin(theta),-1), SE);
[/asy]
(C)
Level 2
Precalculus
Problem: For a constant $c,$ in cylindrical coordinates $(r,\theta,z),$ find the shape described by the equation
\[\theta = c.\](A) Line
(B) Circle
(C) Plane
(D) Sphere
(E) Cylinder
(F) Cone

Enter the letter of the correct option.
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>The equation $\theta = c$ specifies a constant angle in cylindrical coordinates. In cylindrical coordinates, the variable $\theta$ represents the angular position around the z-axis. This means that for any fixed value of $\theta$, the radius $r$ and height $z$ can vary freely. 

To understand the shape described by this equation, consider the implications of varying $r$ and $z$. The radius $r$ can take any non-negative value, meaning it can extend outward from the origin in a circular manner. The height $z$ can also take any value, allowing for vertical movement. 

As a result, the combination of all possible values of $r$ and $z$ while keeping $\theta$ constant creates a surface that extends infinitely in the radial direction and vertically along the z-axis. This surface is a vertical plane that intersects the z-axis and is oriented at the angle $c$ with respect to the positive x-axis in the x-y plane.

Thus, the shape described by the equation $\theta = c$ is a vertical plane. The final answer is $(C)$.<|END_OF_LATENT|> The final answer is $(C)$.