Sorry I wasn't viewing the forum during the summer camp. 

7.8: we can consider the cosine part as the real part, and sine part as the imaginary part, and combine into a complex number, and then it becomes a sequence problem.  For the sum of a sequence whose terms are the products of terms of an arithmetic and a geometric sequence, there is a special technique for it.

7.9: We have talked about this type of questions using cube root of unity.

7.10: This is similar to Example 20.

7.16: Use a complex number to represent each of the points A, B, C, and P, and the lengths can all be expressed as the moduli of differences between those numbers. The inequality come from a triangle inequality; it is complicated, but try to see if you can find a way.