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Translate the equation x + y = 2x2 into polar coordinates.
Replace x with r cos θ and y with r sin θ.
x + y = 2x2
r cos θ + r sin θ = 2 (r cos θ)2 r cos θ + r sin θ = 2r2 cos2θ
One solution to this equation is r = 0. If r isn't zero, we can divide both sides of the equation by r to find
cos θ + sin θ = 2r cos2θ.
Since polar equations are usually written in the form
r = f(θ)
we rearrange this a bit more to find
Translate the inequality x2 + y2 > 1 into polar coordinates.
We can replace x2 + y2withr2 to find the polar inequality r > 1. This describes all points that lie outside the unit circle:
Translate the equation
r cos θ + r sin θ = 2
into rectangular coordinates.
This requires a straightforward substitution.
If r cos θ + r sin θ = 2 then
x + y = 2.
Translate the inequality
into rectangular coordinates, assuming .
The cos θ piece is difficult to translate, but if we multiply both sides of the inequality by cos θ, we find
r cos θ > 3
so x > 3.
The restriction on the values of θ ensures that cosθ isn't negative, this way we don't accidentally multiply the inequality by a negative number.
To go from polar to rectangular coordinates we need to rearrange the equation or inequality until we have terms that can be replaced with x, y, or x2 + y2. We may need to multiply both sides of an equation or inequality by r or by cos θ . We may need to square both sides. We may need to do other things.