%5E2%2By%5E2%3D9-rectangular-to-polar.jpeg "(x-3)^2+y^2=9 Rectangular To Polar")
Converting Rectangular Equation to Polar Form
The equation $(x-3)^2 + y^2 = 9$ represents a circle in rectangular coordinates. Let's convert it to polar coordinates.
Understanding the Conversion Formulas
We can convert from rectangular coordinates $(x, y)$ to polar coordinates $(r, \theta)$ using the following relationships:
- x = r cos(θ)
- y = r sin(θ)
Substituting and Simplifying
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Substitute: Begin by substituting the polar coordinate expressions for x and y into the original equation:
(r cos(θ) - 3)^2 + (r sin(θ))^2 = 9
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Expand: Expand the square and simplify:
r^2 cos^2(θ) - 6r cos(θ) + 9 + r^2 sin^2(θ) = 9
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Simplify: Notice that cos^2(θ) + sin^2(θ) = 1. Combine like terms:
r^2 - 6r cos(θ) = 0
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Solve for r: Factor out r:
r(r - 6 cos(θ)) = 0
This gives us two possible solutions:
- r = 0
- r - 6 cos(θ) = 0 => r = 6 cos(θ)
Interpreting the Solution
- r = 0: This represents the origin, which is a single point on the circle.
- r = 6 cos(θ): This equation defines the entire circle in polar coordinates.
Therefore, the polar form of the equation $(x-3)^2 + y^2 = 9$ is r = 6 cos(θ). This form highlights the circle's symmetry and its dependence on the angle θ.