Solving \(\sin x=k\) over \(0^\circ\le x\le 360^\circ\) gives the solutions in one cycle. The graph repeats the same values forever in both directions, so once no interval is stated the equation has infinitely many solutions. A general solution collects every one of them in a single formula with an integer parameter \(n\).
The same two intersections in every cycle: \(y=\sin x\) and \(y=k\)
The line \(y=k\) meets the curve at the same two angles in every cycle. Stepping one full period left or right from any solution lands on another solution, and the stepping never ends — so each solution found in one cycle generates a whole family of solutions.
Period of each trigonometric function
| Function | Period (degrees) | Period (radians) |
|---|---|---|
| \(\sin x\) | \(360^\circ\) | \(2\pi\) |
| \(\cos x\) | \(360^\circ\) | \(2\pi\) |
| \(\tan x\) | \(180^\circ\) | \(\pi\) |
General-solution patterns for basic angle \(\alpha\) (shown for \(k>0\)), \(n\in\mathbb Z\)
| Equation | General solution (degrees) | General solution (radians) |
|---|---|---|
| \(\sin x=k\) | \(x=\alpha+360^\circ n\) \(x=180^\circ-\alpha+360^\circ n\) | \(x=\alpha+2\pi n\) \(x=\pi-\alpha+2\pi n\) |
| \(\cos x=k\) | \(x=\alpha+360^\circ n\) \(x=360^\circ-\alpha+360^\circ n\) | \(x=\alpha+2\pi n\) \(x=2\pi-\alpha+2\pi n\) |
| \(\tan x=k\) | \(x=\alpha+180^\circ n\) | \(x=\alpha+\pi n\) |
Each sine and cosine row is the pair of quadrant solutions from one cycle, with whole periods \(360^\circ n\) added to each. When \(k\) is negative the quadrant rule picks different angles in the first cycle, but the recipe is unchanged: solve one cycle first, then add whole periods. Tangent needs only one family because its two quadrant solutions \(\alpha\) and \(180^\circ+\alpha\) already differ by exactly one period.
General solutions are extension material. When an examination question states a finite interval, list only the solutions inside that interval — never leave \(+360^\circ n\) in the final answer.
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