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Unformatted text preview: More speciﬁcally, it is the hyperbola obtained
by rotating the graph of x2 − y 2 = 4 counter-clockwise through a 45◦ angle. Armed with polar
coordinates, we can generalize the process of rotating axes as shown below. 11.6.1 Rotation of Axes Consider the x- and y -axes below along with the dashed x - and y -axes obtained by rotating the xand y -axes counter-clockwise through an angle θ and consider the point P (x, y ). The coordinates
(x, y ) are rectangular coordinates and are based on the x- and y -axes. Suppose we wished to ﬁnd
rectangular coordinates based on the x - and y -axes. That is, we wish to determine P (x , y ). While
this seems like a formidable challenge, it is nearly trivial if we use polar coordinates. Consider the
angle φ whose initial side is the positive x -axis and whose terminal side contains the point P .
y P (x, y ) = P (x , y ) x θ
x We relate P (x, y ) and P (x , y ) by converting them to polar coordinates. Converting P (x, y ) to
polar coordinates with r > 0 yields x = r cos(θ + φ) and...
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