Inner Products and Angles of Complex Numbers

One can prove the following propositions: (1) For all real numbers a, b holds −(a + bi) = −a + (−b)i. (2) For all real numbers a, b such that b > 0 there exists a real number r such that r = b · −⌊ b ⌋ + a and 0 ¬ r and r < b. (3) Let a, b, c be real numbers. Suppose a > 0 and b ­ 0 and c ­ 0 and b < a and c < a. Let i be an integer. If b = c + a · i, then b = c. (4) For all real numbers a, b holds sin(a− b) = sin a · cos b− cos a · sin b and cos(a − b) = cos a · cos b + sin a · sin b. (5) For every real number a holds sin(a − π) = −sin(a) and cos(a − π) = −cos(a). (6) For every real number a holds sin(a − π) = −sin a and cos(a − π) = −cos a.