Isolate the ratio. For \(\sin x=k\) or \(\cos x=k\), require \(|k|\le1\).
Find the reference angle from \(|k|\), then use the sign to choose quadrants. Treat values \(0,\pm1\) using the axes.
| Quadrant | Ratio positive there | Solution (degrees) | Solution (radians) |
|---|---|---|---|
| I | |||
| II | |||
| III | |||
| IV |
Reading intersections
\(\sin x=\frac12\), \(0\le x\le2\pi\), gives \(x=\frac\pi6,\frac{5\pi}6\).
Axis values
\(y=\sin x\), \(0\le x\le 2\pi\): peak at \(\dfrac{\pi}{2}\), zeros at \(0\), \(\pi\), \(2\pi\), trough at \(\dfrac{3\pi}{2}\).
\(y=\cos x\), \(0\le x\le 2\pi\): peaks at \(0\) and \(2\pi\), zeros at \(\dfrac{\pi}{2}\), \(\dfrac{3\pi}{2}\), trough at \(\pi\).
| Equation | Solutions for \(0^\circ\le x\le 360^\circ\) | Solutions for \(0\le x\le 2\pi\) |
|---|---|---|
| \(\sin x=1\) | ||
| \(\sin x=0\) | ||
| \(\sin x=-1\) | ||
| \(\cos x=1\) | ||
| \(\cos x=0\) | ||
| \(\cos x=-1\) |
Use the correct angle mode, include allowed endpoints and round only the final answers. The calculator gives only the principal value.
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