Polar Geometry Fundamentals

Exercises

  1. Turn a piece of paper sideways (landscape orientation) and draw a large pair of \(xy\)-coordinate axes. For each of the following polar coordinates \(\bigl(r,\theta\bigr)\) with \(r\) measured in inches (″) and \(\theta\) measured in degrees (°), use a ruler/protractor to accurately plot the location of each point and label the point with its polar coordinates. Once you’ve done this, convert the polar coordinates of each point to rectangular coordinates \((x,y),\) and measure the rectangular coordinates of your plotted point to check the accuracy of its location.
    \(\bigl(2, 30°\bigr)\)
    \(\bigl(4, 10°\bigr)\)
    \(\bigl(2, 111°\bigr)\)
    \(\bigl(1.3, -72°\bigr)\)
    \(\bigl(2, 222°\bigr)\)
    \(\bigl(-5, 166°\bigr)\)
  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 positive \(x\)-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 \(\bigl(r, \theta\bigr)\) with \({r \geq 0}\) and \({0 \leq \theta \lt 360°}\) that describes the same point.
    \(\bigl(3,543°\bigr)\)
    \(\bigl(7,1000°\bigr)\)
    \(\bigl(2,-34°\bigr)\)
    \(\bigl(-7,11°\bigr)\)
    \(\bigl(-5,-4321°\bigr)\)
  6. What is the slope of the line that passes through the origin and the point on the unit circle with coordinates \(\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)?\) At what angle is this line inclined from the positive \(x\)-axis?
  7. What is an equation of the line that passes through the point \((6,7),\) has a positive slope, and forms an acute angle of \(39°\) with the \(x\)-axis?
  8. Consider a sector of a circle of radius 7′ centered at the origin \(\bigl(0,0\bigr)\) with a central angle of 37°, and suppose one edge (radius) of the sector is aligned with the positive \(x\)-axis. What are the coordinates of the terminal point of the other edge?
  9. Consider a sector of a circle of radius 1′ centered at the origin \(\bigl(0,0\bigr)\) that has an outer arc of length 2′, and suppose one edge (radius) of the sector is aligned with the positive \(x\)-axis. What are the coordinates of the terminal point of the other edge?
  10. Consider a sector of a circle of radius 3′ centered at the origin \(\bigl(0,0\bigr)\) that itself has an area of 7 ft², and suppose one edge (radius) of the sector is aligned with the positive \(x\)-axis. What are the coordinates of the terminal point of the other edge?
  11. Explain, as if explaining to a thirteen-year-old, the idea of a rectangular coordinate system vs a polar coordinate system. Feel free to use examples, or to draw any diagrams to help your explanation. It’d be a good idea to include use-cases for each coordinate system in your explanation.

Problems & Challenges

  1. A standard piece of letter paper is 8½″×11″. Suppose you draw a line from the bottom left-hand-corner to the top right-hand corner of such a piece of paper. What angle will that line make with the bottom edge of the paper?
  2. A rain gutter should be installed at a slight incline, usually called the pitch of the gutter. But “pitch” is just another word for “grade” or “slope”. A common recommendation is that the gutter should drop by ¼″ for every 10′ of roofing. What angle does this recommendation correspond to?
  3. You may have seen road signs like the one here that indicate the road is declined steeply at a specific grade. For example, an 8% grade (decline) indicates that there is an 8% loss in elevation per horizontal distance travelled — e.g. for every 100′ a truck travels horizontally it’ll lose 8′ of elevation. This corresponds directly with the “slope” of the road. Considering the surface of the road as a line, the slope of that line will be -8%.
    1. What angle of declination (depression) does a 6% grade correspond to?
    2. If a road is constructed to have a 6° decline, what is the grade of the road?
    3. If a truck drives along a road with a 6° decline for one mile, how much elevation does it lose?
  4. Patroclus the Pirate is following a map to buried treasure. Upon landing on the island at the small dilapidated dock indicated on the map, he notices that there’s no mark at the location of the treasure, but instead just this written instruction: Walk 1234 paces* northwest-by-west, and with my treasure you’ll be blessed. Wary of just pacing it out, Patroclus wants to mark the destination on his map first. How many feet north and how many feet west of the dock is the buried treasure? (Patroclus assumes that a pace refers to a roman pace, which is approximately 5 feet.)
  5. What is the smallest positive real number \(x\) for which the value of \(\cos(x)\) doesn’t depend on whether \(x\) is regarded as the degree measure of an angle or the radian measure of an angle.
  6. What is the measure of the acute angle formed at the intersection of the line \(y = \frac{3}{5}x+7\) and the \(x\)-axis? What is the measure of the acute angle formed at the intersection of the lines \(y = \frac{3}{5}x+7\) and \(y = \frac{7}{2}x-3?\) In general, given two lines \(y = m_1x+b_1\) and \(y = m_2x+b_2,\) what is a formula in terms of \(m_1\) and \(m_2\) and \(b_1\) and \(b_2\) for the measure of the angle at which they intersect?
  7. In rectangular coordinates, \(y = mx+b\) is an equation for a line with slope \(m\) and \(y\)-intercept \(b.\) How do you express the equation for a line in polar coordinates? I.e. an equation relating coordinates \(r\) and \(\theta\) instead of \(x\) and \(y?\) Solving this equation for \(r\) figure out which values of \(\theta\) will correspond to negative values of \(r.\)
  8. Roberta has just designed a small walking robot and has taken it out to a soccer field to test it out. This field’s long edge runs east-to-west and short edge runs north-to-south, and the field’s dimensions adhere precisely to FIFA’s recommendation of 105×68 meters. Roberta places the robot at one corner of the field and gives it the instruction to walk to the diagonally opposite corner along the shortest path possible. However, Roberta doesn’t realize that her robot’s navigation system has a design flaw: it can only travel in the directions on a 32-point compass, along the principal, half-, and quarter-winds. The robot still manages to determine the shortest path possible under this constraint and walks to the opposite corner, albeit its path isn’t a straight line. What path did the robot take, and how far was its walk?