In regard to the use of trigonometric functions such as the Sine and Cosine of a particular angle, a simple scientific hand calculator can be used to evaluate these functions. However, I have provided the table below which provides this information. Note that a property of the Sines and Cosines is that sum of their squares are equal to 1.
In Lesson 4, it is stated that the maximum L/D is attained at the angle-of- attack where the parasite drag and induced drag are equal. This conclusion is found by using elementary calculus when trying to locate the maximum of a function. However, it is easy to prove this fact by picking any two values of CD0 and k and plotting the L/D ratio versus the lift coefficient. This can be done using Microsoft Excel. When the maximum is located, you will observe that the parasite drag and the induced drag coefficients are equal.
In the case of a C-172 with the propeller windimilling, the value of CD0 is approximately 0.05, and the value of k is approximately 0.06. These values would produce a maximum L/D of 9.09.