The Trajectory of a Point Particle in Polar Coordinates

To determine the trajectory of a point particle in the (x, y) plane given in polar coordinates, we start by converting the equation from polar to Cartesian coordinates.

1. Converting the Equation to Cartesian Coordinates
The given equation is r sin(theta * pi/4) = 5, where r represents the radial distance and theta represents the angle in polar coordinates. To convert this equation to Cartesian coordinates, we use the following relationships:

x = r cos(theta)
y = r sin(theta)

Substituting these values into the given equation, we have:
r sin(theta * pi/4) = 5
r * sin(theta * pi/4) = 5

Using the relationship y = r sin(theta), we can rewrite the equation as:
y = 5

This implies that the y-coordinate of the particle is always 5, regardless of the value of x. Hence, the trajectory of the particle is a straight line parallel to the x-axis.

2. Explanation
The equation y = 5 represents a horizontal line in the (x, y) plane, with y-coordinate always equal to 5. This means that regardless of the angle theta or the radial distance r, the particle will always have a y-coordinate of 5.

Since the equation of the trajectory does not include any terms involving x, the x-coordinate of the particle is not affected by the angle theta or the radial distance r. Therefore, the particle will move horizontally along the line y = 5, parallel to the x-axis.

Conclusion
The trajectory of the point particle described by the equation r sin(theta * pi/4) = 5 in polar coordinates is a straight line parallel to the x-axis. The equation y = 5 represents a horizontal line in the (x, y) plane, with the y-coordinate always equal to 5. The x-coordinate of the particle remains unchanged, resulting in a straight-line trajectory.