Math - Trigonometry
Time Tutored: 3:30-4:30
Additional Billable Time: N/A
Previous Quiz/Test Scores: N/A
Next Scheduled Session: Sometime next week
Yesterday's session focused on polar coordinates and polar graphs, which is the material that had been covered in class. B seems to understand the basic principle of polar coordinates, which is that coordinates are given as a distance from the origin (r = sqrt(x^2 + y^2)) and the counter-clockwise angle the line drawn to that point from the origin makes with the positive x-axis (tan(theta) = y/x). Additionally, x = r*cos(theta) and y = r*sin(theta) (where x and y are rectangular coordinates). Everything comes from how these coordinates relate to rectangular coordinates. Converting between the two simply means substituting the relations for x and y or r and theta.
Visualizing a graph from a polar equation is a bit more difficult, since it is not immediately obvious what a given relationship between r and theta means. There are a few basic rules to remember, but I don't think that most of them had been covered quite yet. The one thing that was covered is that a graph of the form r = a*sin(n*theta) or r = a*cos(n*theta) will give a rose-like graph with n petals if n is odd and 2n if n is even. This difference arises from the symmetry of trig functions about the point or the vertical line at x = pi. When the function has line symmetry, it has the same value for theta and theta + pi, and since these angles are opposite of each other, this makes the function draw a mirror image of itself in polar graphs; while if it has point symmetry, it has opposite values for theta and theta + pi, which makes the function double back on itself since (-r, theta + pi) = (r, theta). There are a number of other forms of graphs, but they did not appear to have been covered in the textbook sections B highlighted.
The important thing is to remember how to convert between polar and rectangular coordinates. He seems to get this in principle, but sometimes gets a bit tripped up in the algebraic manipulation, which comes with some practice. As for polar graphs, most of it is memorizing a few rules and, if all else fails, a graphing calculator will do the trick. I doubt he'll have to graph anything complicated by hand, so that shouldn't be a major issue. Also, interestingly, as I told him - polar coordinates are very similar to complex numbers since r and theta are defined in virtually the same way for both, if it helps to think in that way.
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