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Let $ \theta = 2z + 60^\circ $, so:
2z + 60^\circ = 30^\circ + 360^\circ k \Rightarrow 2z = -30^\circ + 360^\circ k \Rightarrow z = -15^\circ + 180^\circ k
2z + 60^\circ = 150^\circ + 360^\circ k \Rightarrow 2z = 90^\circ + 360^\circ k \Rightarrow z = 45^\circ + 180^\circ k
Now find all $ z \in [0^\circ, 360^\circ] $:
For $ z = -15^\circ + 180^\circ k $:
$ k = 1 $: $ 165^\circ $
$ k = 2 $: $ 345^\circ $
$ k = 0 $: $ -15^\circ $ — invalid
$ k = 3 $: $ 525^\circ $ — too large
For $ z = 45^\circ + 180^\circ k $: