Set the whole argument equal to a new angle and transform the interval before solving.
Example: \(\sin(2x-30^\circ)=\frac12\), \(0^\circ\le x\le360^\circ\)
\(\theta=2x-30^\circ\qquad-30^\circ\le\theta\le690^\circ\)
| Step | Working |
|---|---|
| Reference angle | |
| Quadrants I and II (sine positive) | |
| Add \(360^\circ\) within \(-30^\circ\le 2x-30^\circ\le 690^\circ\) | |
| Return to \(x\) |
Add full periods inside the transformed interval, then recover \(x\) and check the original bounds. A factor of \(2\) in the argument doubles the number of cycles, but endpoint solutions still need checking.
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