Solving Trigonometric Equations

A trig equation like sin(x) = 1/2 asks: where on the circle does the y-coordinate equal 1/2? That happens at two points every single revolution, so the solutions repeat forever, spaced by the period. That is why one equation has infinitely many answers marching off by multiples of 2pi, or by π for tangent.

To solve one, isolate the trig function first, exactly like isolating a variable. Then apply the matching inverse function, arcsin, arccos, or arctan, to get the principal solution, the one value inverse trig functions are built to hand you. From there, use the unit circle to find any other angle in one revolution with the same sine, cosine, or tangent value, usually a reference-angle partner. Finally add the period, 2pi·k for sine, cosine, and most equations, or π·k for tangent, to sweep out every solution, where k is any integer.

When the equation has an inner argument, like cos(2x) = -1/2, solve for the whole argument first, then divide by the coefficient last so the period divides too: 2x lands every 2pi, but x lands every π. When a problem restricts the domain to a real window, like time over one cycle of a Ferris wheel, you stop generating solutions once k pushes x outside that window, keeping only the physically meaningful answers.