In Appendix D of the 6th edition of Stewart's Calculus, we find example 3 which tells us to find the exact trigonometric ratios for $t = 2\pi/3$. The solution says --- from Figure 10 we see that a point on the terminal line for $t = 2\pi/3$ is $P(-1, \sqrt 3)$. From this point I can follow the example. After solving the problem, Stewart shows a table of various values for $\sin t$ and $\cos t$ saying these were computed from the method of Example 3.
What I cannot do alone is come up with Figure 10.
(Figure 10, Appendix D, Stewart's Calculus, 6th edition)
I know how to draw the angles. I can find the point (-1, 0) and deduce that base of the triangle has length 1. I know I can stretch the angle-delimiting line and say it has length 2. Omg, I just solved my difficulty: now with Pythagoras I can get the height $\sqrt 3$ and then I will have the point $P$. Until now I had not realized how he had reached the coordinates of the point $P$.
Okay, so let me ask another question --- are there any other methods of building these tables? I need to reduce the amount of information I must memorize. I need methods that I can recall quickly and compute quickly on the spot --- on a test, say.
I once read the chapter on trigonometry by Alfred North Whitehead's "An Introduction to Mathematics". I'm going to read it again to see if he teaches us to build such tables. Any other recommendation?