You might find it useful to use the Unit Circle GSP sketch while reflecting on this discussion in 1-4.

  1. What is a degree?
    1. An arc length that is 1/360 of a circle's circumference
  2. Look at a unit circle. What, approximately is the length of sin(40°), in degrees?
    1. sin(40°) is approximately 0.64, which is 64% of one radius, so sin(40°) is 0.64 radians. One radian is 360/(2π) degrees, so 0.64 radians is approximately 36°.
  3. What, then, is cos(sin(40°)?
    1. It is cos(36°), which is approximately 0.80.
  4. Set your calculator into "degree" mode. On your calculator:
    1. Calculate sin(40°). [0.642788]
    2. What does 0.642788 stand for? [A percent of one radius]
    3. Calculate cos(0.642788). [i.e., cos(sin(40°)].
    4. You get 0.9999. Why? [Because the calculator, being in degree mode, assumed that 0.642788 was a number of degrees, when in fact it was a number of radians.]
    5. Now calculate cos(sin(40°)). What do you get? You get 0.80. Why? Because calculator manufacturers program their calculators so that when you enter a trig function that has a function as its argument, the calculator automatically interprets the argument's output as a number of radians.

We are accustomed to thinking that if a function is linear, then it has a line as its graph, and that if a function has a line as its graph, then it is linear. In this discussion, we raise the issue of what is "linearness" that makes a function a linear function.

  1. In what ways are f(x) = 5 - 2x and f(θ) = 5 - 2θ both linear? Do they both have lines as their graphs?
  2. Is the graph of r = 2 a line? Why or why not?
  3. Is the graph of r = 1/sin(θ) a line? Is it a line in polar coordinates? Is it a line in rectangular coordinates? Why or why not?
  4. Is the graph of θ = 3 a line? Why or why not?
  5. What is the central issue being raised here?
  6. What is the property of a graph, independent of the coordinate system, that makes it a "line" within that system?