to answer questions like these which involve both a

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Unformatted text preview: r = 3 cos 2 . Using the techniques presented in Example 11.5.2, we find that we need to plot both functions as θ ranges from 0 to 4π to obtain the complete graph. To our surprise and/or delight, it appears as if these two equations describe the same curve! y 3 −3 3 x −3 θ r = 3 sin 2 and r = 3 cos θ 2 appear to determine the same curve in the xy -plane To verify this incredible claim,16 we need to show that, in fact, the graphs of these two equations intersect at all points on the plane. Suppose P has a representation (r, θ) which θ θ satisfies both r = 3 sin 2 and r = 3 cos 2 . Equating these two expressions for r gives θ θ the equation 3 sin 2 = 3 cos 2 . While normally we discourage dividing by a variable θ expression (in case it could be 0), we note here that if 3 cos 2 = 0, then for our equation θ to hold, 3 sin 2 = 0 as well. Since no angles have both cosine and sine equal to zero, θ θ θ we are safe to divide both sides of the equation 3 sin 2 = 3 cos 2 by 3 cos 2 to get θ tan 2 = 1 which gives θ = π + 2πk for integers k . From these solutions, however, we 2 14 Again, we could have easily chosen to substitute these into r = 3 which would give −r = 3, or r = −3. We obtain t...
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This note was uploaded on 05/03/2013 for the course MATH Algebra taught by Professor Wong during the Fall '13 term at Chicago Academy High School.

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