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

- What is a degree?
*An arc length that is 1/360 of a circle's circumference*

- Look at a unit circle. What, approximately is the length of sin(40°), in degrees?
*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°.*

- What, then, is cos(sin(40°)?
*It is cos(36°), which is approximately 0.80.*

- Set your calculator into "degree" mode. On your calculator:
- Calculate sin(40°). [0.642788]
- What does 0.642788 stand for? [
*A percent of one radius*] - Calculate cos(0.642788). [
*i.e., cos(sin(40*°*)*]. - 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.*] - 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.

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