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Unformatted text preview: , we take a step back and think geometrically.
We know y = −x describes a line through the origin. As before, r = 0 describes the
origin, but nothing else. Consider the equation θ = − π . In this equation, the variable
r is free,8 meaning it can assume any and all values including r = 0. If we imagine
plotting points (r, − π ) for all conceivable values of r (positive, negative and zero), we
are essentially drawing the line containing the terminal side of θ = − π which is none
other than y = −x. Hence, we can take as our ﬁnal answer θ = − π here.9
(c) We substitute x = r cos(θ) and y = r sin(θ) into y = x2 and get r sin(θ) = (r cos(θ))2 , or
r2 cos2 (θ) − r sin(θ) = 0. Factoring, we get r(r cos2 (θ) − sin(θ)) = 0 so that either r = 0
or r cos2 (θ) = sin(θ). We can solve the latter equation for r by dividing both sides of
the equation by cos2 (θ), but as a general rule, we never divide through by a quantity
that may be 0. In this particular case, we are safe since...
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