Unformatted text preview: tions, we assigned to each angle a position on the Unit Circle. In
this subsection, we broaden our scope to include circles of radius r centered at the origin. Consider
for the moment the acute angle θ drawn below in standard position. Let Q(x, y ) be the point on
the terminal side of θ which lies on the circle x2 + y 2 = r2 , and let P (x , y ) be the point on the
terminal side of θ which lies on the Unit Circle. Now consider dropping perpendiculars from P and
Q to create two right triangles, ∆OP A and ∆OQB . These triangles are similar, 10 thus it follows
that x = 1 = r, so x = rx and, similarly, we ﬁnd y = ry . Since, by deﬁnition, x = cos(θ) and
y = sin(θ), we get the coordinates of Q to be x = r cos(θ) and y = r sin(θ). By reﬂecting these
points through the x-axis, y -axis and origin, we obtain the result for all non-quadrantal angles θ,
and we leave it to the reader to verify these formulas hold for the quadrantal angles.
y y r Q (x, y )
Q(x, y ) = (r cos(θ), r sin(θ))
1 P (x , y )
1 P (x ,...
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