Angles on the unit circle and complete turns (II)

Statement

Represent the following angles on the unit circle, also showing the number of complete turns made, if any.

• (a) $-123^\circ$
• (b) $1643^\circ$
• (c) $-500^\circ$
• (d) $800^\circ$

Solution

Reduce each angle to the interval $[0^\circ,360^\circ)$ by removing (or adding) multiples of $360^\circ$; the number of multiples is the number of complete turns.

• (a) $-123^\circ$: principal coterminal angle $360^\circ-123^\circ=237^\circ$; no complete turn ($0$).
• (b) $1643^\circ-4\cdot360^\circ=1643^\circ-1440^\circ=203^\circ$; $4$ complete turns (counterclockwise).
• (c) $-500^\circ+2\cdot360^\circ=220^\circ$; this corresponds to $1$ complete clockwise turn ($-500^\circ=-360^\circ-140^\circ$).
• (d) $800^\circ-2\cdot360^\circ=80^\circ$; $2$ complete turns (counterclockwise).

$\boxed{237^\circ\,(0);\quad 203^\circ\,(4);\quad 220^\circ\,(1);\quad 80^\circ\,(2)}$