Every Solution

With no domain, an equation has infinitely many answers, and you write them all at once. sin(x) = √2/2 has 45° and 135° in one turn, and every full turn adds the same two again. So every solution is

x = 45° + 360°·k

x = 135° + 360°·k

where k is any integer: …, −2, −1, 0, 1, 2, …. k = 1 gives 405° and 495°; k = −1 gives −315° and −225°. Each line is called a family. Find the answers in one turn, then add 360°k to each.