Lessons of the crosswalk

• Ð in radians, sin Ð, and tan Ð are almost equal (.03491, .03490, .03492).
• cos Ð is very close to 1 (even closer than the other functions are to 0 (.99939).
• 1 - cos Ð is somewhat of a nuisance to get to 2 decimal places (.00061). (If you didn't find it a nuisance, try it with a smaller angle, such as .00001 radians.)
• For practical purposes, there isn't much point in calculating sin and tan when Ð is small, since they are essentially equal to Ð.
• In other words, the graphs of sin and tan are well approximated near Ð = 0 by a straight line (the linear approximation, Sec. 2.9, Class 17.T).
• A worthwhile question to ask is: How large can Ð be before this approximation can no longer be trusted?
• For cos and 1 - cos, the approximating straight line is horizontal. Therefore, to get nontrivial approximations to them we need to use the quadratic approximation (also discussed in Sec. 2.9).
• In other words, the graph of cos or 1 - cos is well approximated near Ð = 0 by a parabola.