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Unformatted text preview: y 2 = x2 + y 2 + x . Once again, we have performed some
10 Exercise 3 in Section 5.3, for instance . . .
Here, ‘equivalent’ means they represent the same point in the plane. As ordered pairs, (3, 0) and (−3, π ) are
diﬀerent, but when interpreted as polar coordinates, they correspond to the same point in the plane. Mathematically
speaking, relations are sets of ordered pairs, so the equations r2 = 9 and r = −3 represent diﬀerent relations since
they correspond to diﬀerent sets of ordered pairs. Since polar coordinates were deﬁned geometrically to describe the
location of points in the plane, however, we concern ourselves only with ensuring that the sets of points in the plane
generated by two equations are the same. This was not an issue, by the way, when we ﬁrst deﬁned relations as sets
of points in the plane in Section 1.2. Back then, a point in the plane was identiﬁed with a unique ordered pair given
by its Cartesian coordinates.
In addition to taking the tangent of both sides of an equation (There are inﬁni...
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