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Unformatted text preview: θ) = 0
−A sin(2θ) + B cos(2θ) + C sin(2θ) = 0 Double Angle Identities
From this, we get B cos(2θ) = (A − C ) sin(2θ), and our goal is to solve for θ in terms of the
coeﬃcients A, B and C . Since we are assuming B = 0, we can divide both sides of this equation
by B . To solve for θ we would like to divide both sides of the equation by sin(2θ), provided of
course that we have assurances that sin(2θ) = 0. If sin(2θ) = 0, then we would have B cos(2θ) = 0,
and since B = 0, this would force cos(2θ) = 0. Since no angle θ can have both sin(2θ) = 0 and
cos(2θ) = 0, we can safely assume sin(2θ) = 0.3 We get cos(2θ) = A−C , or cot(2θ) = A−C . We have
just proved the following theorem.
Theorem 11.10. The equation Ax2 + Bxy + Cy 2 + Dx + Ey + F = 0 with B = 0 can be
transformed to an equation in variables x and y without any x y terms by rotating the x and y
axes counter-clockwise through an angle θ which satisﬁes cot(2θ) = A−C .
We put Theorem 11.10 to good use in the following example.
Example 11.6.2. Graph the followin...
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