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I have worked out the answers $45^\circ$, $135^\circ$, $225^\circ$, $315^\circ$; although some of these may be wrong. Where have I gone wrong (if I have)? And are there any other answers (between $0^\circ$ and $360^\circ$ inclusive) that I have missed out?

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$\sin30^\circ = \sin150^\circ = \sin(30^\circ+360^\circ) = \sin(150^\circ+360^\circ) = \dfrac{1}{2}$ so you need to seek $x$ values where

$$2x - 60 = 30 \implies x = 45^\circ$$

$$2x-60 = 150 \implies x = 105^\circ$$

But notice that since $x \in [0^\circ,360^\circ]$, we also may have

$$2x-60 = 390 \implies x = 225^\circ$$

$$2x-60 = 510 \implies x = 285^\circ$$

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Note that $$\sin y=\frac12\iff y=30°+2k\cdot180°,150°+2k\cdot180°\implies 2x=90°+k360°,210°+k360°$$

thus since $x\in[0,360°]$

$$x=45°,225°$$

$$x=105°,285°$$

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  • $\begingroup$ sorry for the initial confusion, now it should be ok! $\endgroup$
    – user
    Feb 8 '18 at 21:32

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