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    Polar Coordinates (25 Problems)

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    Please do problems 1,2,3,4,5,6,7,9,11,15,17,19,18,21,23,25,22,26,29-45 odd, 55,57,61, and 65.
    Please see the attached file for the fully formatted problems.

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    Solution Preview

    In problems 1-4 you are asked to PLOT points that are given in Polar Coordinates. Polar coordinates are given in the form ( r, theta ), where r is the radius and theta is the angle in radians, measured counterclockwise about the origin from the right branch of the horizontal axis.

    For each of these problems (which each have three parts), draw a separate graph for each of the parts. Use graph paper. Draw a horizontal axis and a vertical axis that gross each other at the origin of your plot. You now have to scale your axes. Due to the typical values of r used in these problems (i.e., absolute value of 3 or less) each of the four axis branches should be ticked off with four major divisions (numbered 1, 2, 3, 4, outwards from the origin), depending on the grid size of your graph paper, you may want to use 5 or 10 grid squares within each of these major divisions. These labels must be negative on the downward and leftward branches.

    Measure the given angle theta about the origin (i.e., the point of intersection of the horizontal and vertical axes) counterclockwise from the right branch of the horizontal axis using a protractor. If your protractor only reads in degrees, convert radians into degrees by multiplying by 180 and dividing by pi (i.e., pi = 3.14156). Make a small mark on the paper next to the measured angle at the edge of the protractor. The point you want to plot will lie on a line through this point and the origin.

    Special cases: If theta is negative then take absolute value first, but measure the angle CLOCKWISE instead. If the absolute value of theta is GREATER than pi (i.e., greater than 180 degrees), then subtract pi from it first, but measure from the LEFT branch of the horizontal axis in that case.

    The next step in plotting the point in polar coordinates is to draw a thin line from the origin through the small mark you made next to the protractor, and past it a bit, using a straight edge. Also extend it the same distance on the opposite side of the origin.

    Then take a compass and lightly press the sharp end at the origin of your plot. Adjust the free arm of the compass so that the tip of the pencil point lies on the right branch of the horizontal axis, at a distance from the origin equal to the absolute value of r. Swing the compass so that an arc is drawn (lightly) on the paper from the right branch of the horizontal axis to the line drawn earlier. Draw the arc to the side of the line that is on the same side of the origin as the mark you made with the protractor. If r is positive you can place a bold dot at this point, and label it as (r=__ , theta=__ ), where the underlines are replaced with the actual values. If r is negative, swing the pencil point of the compass around to the OPPOSITE side of the previous line drawn, without continuing the first arc, then tick that line with a short arc, and place the bold point at that intersection, labelled as explained above.

    In problems 1 and 2, an additional step is needed. Two additional polar coordinate representations must be calculated for each of these points, one with r > 0, and one with r < 0. The first of these two additional points is calculated by reversing the sign of r (from + to -, or vice-versa) and at the same time adding pi to theta (or subracting pi from theta, if that gives a simpler result). The second of these two additional points is calculated by leaving r unchanged, but subtracting 2pi from theta (i.e., where 2pi mean two times pi) if theta is positive, or adding 2pi to theta in the case where it is already negative.

    You may wish to check your work by attempting to plot the point given by each of the newly calculated polar coordinates and observe that it does indeed take you back to the same point already plotted in every case.

    Problems 3 and 4 also have an extra step, this is to convert the polar coordinates into Cartesian coordinates. This can be done using the conversion formula:

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

    This means to calculate the sine or cosine of the angle theta, using either a calculator or a table of function values for special angles, then multiply the result by the value of r. Don't forget to carry the proper sign of both ...

    Solution Summary

    Polar coordinates are investigated.