Stitz-Zeager_College_Algebra_e-book

Now that the product rule has been established we use

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Unformatted text preview: ement. −r ? = (−r )2 + (−r cos(θ + π )) = (r )2 − r cos(θ + π ) ? Since r = −(r )2 − r cos(θ ) ? − −(r )2 − r cos(θ ) Since cos(θ + π ) = − cos(θ ) (r )2 + r cos(θ ) = (r )2 − r (− cos(θ )) (r )2 + r cos(θ ) = (r )2 + r cos(θ ) Since both sides worked out to be equal, (−r , θ + π ) satisfies r = r2 + r cos(θ) which 2 means that any point (r, θ) which satisfies r2 = r2 + r cos(θ) has a representation which satisfies r = r2 + r cos(θ), and we are done. In practice, much of the pedantic verification of the equivalence of equations in Example 11.4.3 is left unsaid. Indeed, in most textbooks, squaring equations like r = −3 to arrive at r2 = 9 happens without a second thought. Your instructor will ultimately decide how much, if any, justification is warranted. If you take anything away from Example 11.4.3, it should be that relatively nice things in rectangular coordinates, such as y = x2 , can turn ugly in polar coordinates, and vice-versa. In the next section, we devote our attention to graphing equ...
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