"One way to do this is as follows."
"If" \ \ z \ \ "is a complex number, then:"
\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad z \ = \ r ( cos theta + i sin theta ); \qquad \quad "where:"
\quad r = "magnitude (length) of" \ \ z,
\quad \theta = "the angle" \ \ z \ \ "makes, in radians when" \ z \ "is placed in"
\qquad \qquad \qquad "standard position in the complex plane."
"Now visualize" \ z = -i \ \ "on the complex plane. It lies on the"
"negative" \ x "-axis, 1 unit in length, to the left of the origin. The"
"angle of" \ -i \ \ "in this, its standard position, is clearly" \ \ \pi \ \"radians."
"So now we see, for" \ z = -i :
\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad r = 1,
\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad \theta = \pi.
"Thus:"
\qquad \qquad \qquad \qquad \qquad \qquad \quad -i \ = \ 1 cdot ( cos \pi + i sin \pi )
\qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad \quad \ \ = \ cos \pi + i sin \pi.
"So:"
\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad -i \ = \ cos \pi + i sin \pi.
"This is our answer."