For each solution family \(x=\alpha+Tn\), substitute into the stated interval and keep integer values of \(n\).
\(L\le\alpha+Tn\le U\quad\Longrightarrow\quad\dfrac{L-\alpha}{T}\le n\le\dfrac{U-\alpha}{T}\quad(T>0)\)
Example: \(\sin x=\dfrac1{\sqrt2}\), \(5\pi\le x\le7\pi\)
| Step | Working |
|---|---|
| Basic angle | \(\alpha=\sin^{-1}\dfrac{1}{\sqrt{2}}=\dfrac{\pi}{4}\) |
| General solution | \(x=\dfrac{\pi}{4}+2\pi n\) or \(x=\dfrac{3\pi}{4}+2\pi n\) |
| Solve for \(n\): first family | \(5\pi\le\dfrac{\pi}{4}+2\pi n\le 7\pi\) \(2.375\le n\le 3.375\) \(n=3\) |
| Solve for \(n\): second family | \(5\pi\le\dfrac{3\pi}{4}+2\pi n\le 7\pi\) \(2.125\le n\le 3.125\) \(n=3\) |
| Solutions | \(x=\dfrac{\pi}{4}+6\pi=\dfrac{25\pi}{4}\) \(x=\dfrac{3\pi}{4}+6\pi=\dfrac{27\pi}{4}\) |
Negative integers give negative-angle solutions, e.g. \(\frac\pi4+2\pi(-1)=-\frac{7\pi}4\).
Treat each family separately and check whether the endpoints are included.
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