Angles on the unit circle and complete turns (I)

Statement

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

• (a) $-23^\circ$
• (b) $643^\circ$
• (c) $400^\circ$
• (d) $-1000^\circ$

Solution

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

• (a) $-23^\circ$: principal coterminal angle $360^\circ-23^\circ=337^\circ$; no complete turn ($0$).
• (b) $643^\circ-360^\circ=283^\circ$; $1$ complete turn (counterclockwise).
• (c) $400^\circ-360^\circ=40^\circ$; $1$ complete turn (counterclockwise).
• (d) $-1000^\circ+3\cdot360^\circ=80^\circ$; this corresponds to $2$ complete clockwise turns ($-1000^\circ=-2\cdot360^\circ-280^\circ$).

$\boxed{337^\circ\,(0);\quad 283^\circ\,(1);\quad 40^\circ\,(1);\quad 80^\circ\,(2)}$