Polar Geometry Fundamentals

  1. Turn a piece of paper sideways and draw a large pair of \(xy\)-coordinate axes. For each of the following pairs of polar coordinates \((r,\theta)\) with \(r\) measured in inches, use a ruler/protractor to accurately plot the location of each point, convert the polar coordinates to rectangular coordinates \((x,y),\) and measure the rectangular coordinates of your plotted point to check the accuracy of its location.
    \(\bigl(2, 30°\bigr)\)
    \(\bigl(4, 10°\bigr)\)
    \(\bigl(3, 88°\bigr)\)
    \(\bigl(2, 111°\bigr)\)
    \(\bigl(1.3, -72°\bigr)\)
    \(\bigl(2, 222°\bigr)\)
    \(\bigl(-5, 166°\bigr)\)
  2. Which of the points with rectangular coordinates \((8,15)\) and \((15.5,-7)\) and \((-11,13),\) is closest to the origin?
  3. Which of the points with rectangular coordinates \((9,14)\) and \((27,43)\) and \((21,32),\) is inclined at the smallest angle from the \(x\)-axis?
  4. There is a unique line that passes through the points \((6,-1)\) and \((777,177).\) At what angle is this line inclined? I.e. what is the acute angle that this line makes with the \(x\)-axis?
  5. Each of these pairs polar of coordinates describes a point. Determine the “principal” pair of polar coordinates \((r, \theta)\) with \({r \geq 0}\) and \({0 \leq \theta \lt 360°}\) that describes the same point.
    \(\bigl(3,1000°\bigr)\)
    \(\bigl(2,-34°\bigr)\)
    \(\bigl(-7,11°\bigr)\)
    \(\bigl(-5,-4321°\bigr)\)