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Unformatted text preview: y = r sin(θ + φ). To convert the point
P (x , y ) into polar coordinates, we ﬁrst match the polar axis with the positive x -axis, choose the
same r > 0 (since the origin is the same in both systems) and get x = r cos(φ) and y = r sin(φ).
Using the sum formulas for sine and cosine, we have
x = r cos(θ + φ)
= r cos(θ) cos(φ) − r sin(θ) sin(φ) Sum formula for cosine = (r cos(φ)) cos(θ) − (r sin(φ)) sin(θ)
= x cos(θ) − y sin(θ) Since x = r cos(φ) and y = r sin(φ) 11.6 Hooked on Conics Again 827 Similarly, using the sum formula for sine we get y = x sin(θ) + y cos(θ). These equations enable us
to easily convert points with x y -coordinates back into xy -coordinates. They also enable us to easily
convert equations in the variables x and y into equations in the variables in terms of x and y .1 If
we want equations which enable us to convert points with xy -coordinates into x y -coordinates, we
need to solve the system
x cos(θ) − y sin(θ) = x
x sin(θ) + y cos(θ) = y
for x and y . Perhaps the cle...
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