Question 1137898
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It is unusual to talk about symmetry about the x-axis,  the y-axis, or the origin for a polar graph....<br>
However, the problem can be answered.<br>
To start with, the constants 5 and 3 have no effect on the answer to the problem; so we only need to consider the polar graph of cos(x).<br>
For x-axis symmetry in a polar graph, it means we are replacing the angle x with the angle (360-x).  So the question is whether cos(360-x) is equal to cos(x).  The formula for the cosine of the difference of two angles tells us that it is:<br>
{{{cos(360-x) = cos(360)cos(x)+sin(360)sin(x) = cos(x)}}}<br>
So the graph of r = 5cos(3x) has symmetry about the x-axis.<br>
(NOTE: You can get the same result by viewing x-axis symmetry as meaning the angle x gets replaced by the angle (-x).)<br>
For y-axis symmetry in a polar graph, it means we are replacing the angle x with the angle (180-x).  Is cos(180-x) equal to cos(x)?  No:<br>
{{{cos(180-x) = cos(180)cos(x)+sin(180)sin(x) = -cos(x)}}}<br>
The graph of r = 5cos(3x) does not have symmetry about the y-axis.<br>
And for symmetry about the origin for a polar graph, it means we are replacing the angle x with the angle (x+180).  Is cos(x+180) equal to cos(x)?  No:<br>
{{{cos(x+180) = cos(x)cos(180)-sin(x)sin(180) = -cos(x)}}}<br>
The graph of r = 5cos(3x) does not have symmetry about the origin.<br>
ANSWER: The graph of r = 5cos(3x) has symmetry about the x-axis only.