Final answer:
To plot the polar coordinates (-2, 21pi/4), simplify the angle to determine its position within one full rotation (0 to 2pi), and then plot the point 2 units from the origin in the direction opposite to the simplified angle due to the negative radial coordinate.
Step-by-step explanation:
To plot the polar coordinates (-2, 21pi/4), we must consider the radial coordinate and the angle. The radial coordinate (the distance to the origin in a polar coordinate system) is -2, which implies moving 2 units from the origin towards the opposite direction of the radial line. The angle, given as 21pi/4, needs to be simplified since it is more than a full rotation (which is 2pi radians or 360 degrees). We simplify 21pi/4 by dividing the angle by 2pi to find the equivalent angle within one rotation. Here's how it is done:
- Simplify the angle: 21pi/4 = 21pi/(2*2pi) = 5pi + 1pi/4 = 5*2pi + pi/4, which is equivalent to 5 full rotations plus pi/4.
- Since the radial coordinate is negative, we plot the point in the opposite direction of the angle pi/4, or 45 degrees, measured from the positive x-axis.
- Thus, move 2 units from the origin in the direction 45 degrees from the negative x-axis, because of the negative radial coordinate.
In the context of finding the distance between two points provided in Cartesian coordinates or converting between Cartesian and polar coordinates, one could use the Pythagorean theorem and trigonometric functions to make these calculations.