To analyze the symmetry of the equation given in polar coordinates as r = 9 sin(78), we need to explore the properties of such equations regarding their symmetry about the axes and the origin.
Symmetry About the X-Axis
A graph is symmetric about the x-axis if replacing y with -y results in the same equation. In polar coordinates, this is typically checked using equations related to angles of theta. However, since our equation only contains r (the radius) and a sine function, there isn’t a y-component that we can directly evaluate.
Symmetry About the Y-Axis
A graph is symmetric about the y-axis if replacing x with -x keeps the equation unchanged. In polar coordinates, this again can be trickier. The y-axis symmetry can often be assessed similarly by examining if r = 9 sin(78 +
pihat) holds the same value over a specific range. In our case, replacing theta with -theta leads to the same sine function, suggesting potential symmetry.
Symmetry About the Origin
A graph is symmetric about the origin if replacing (x, y) with (-x, -y) results does not alter the equation. In polar form, this translates to r = 9 sin(78 +
pihat), and since sine retains its values symmetrically, we have an indication of symmetry about the origin as well.
Conclusion
Considering our specific case of r = 9 sin(78), we find that the graph possesses symmetry regarding both the y-axis and the origin. It exhibits unique polar characteristics due to its inherent sine function. However, we cannot conclude about x-axis symmetry based solely on the given polar equation as it doesn’t apply in traditional Cartesian terms.