How to Convert Equations From Rectangular to Polar Form

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In trigonometry, the use of the rectangular (Cartesian) coordinate system is very common when graphing functions or systems of equations. However, under certain conditions, it is more useful to express the functions or equations in the polar coordinate system. Therefore, it may be necessary to learn to convert equations from rectangular to polar form.

    Understand that you represent a point P in the rectangular coordinate system by an ordered pair (x, y). In the polar coordinate system the same point P has coordinates (r, θ) where r is the directed distance from the origin and θ is the angle. Note that in the rectangular coordinate system, the point (x,y) is unique but in the polar coordinate system the point (r, θ ) is not unique (see Resources).

    Know that the conversion formulas that relate the point (x,y) and (r, θ) are: x= rcos θ, y=rsin θ, r²= x² + y² and tan θ= y/x. These are important for any type of conversion between the two forms as well as some trigonometric identities (see Resources).

    Use the formulas in Step 2 to convert the rectangular equation 3x-2y=7 into polar form. Try this example to learn how the process works.

    Substitute x= rcos θ and y=rsin θ into the equation 3x-2y=7 to get (3 rcos θ- 2 rsin θ)=7.

    Factor out the r from the equation in Step 4 and the equation becomes r(3cos θ -2sin θ)=7.

    Solve the equation in Step 5 for r by dividing through both sides of the equation by (3cos θ -2sin θ). You find that r= 7/(3cos θ -2sin θ). This is the polar form of the rectangular equation in Step 3. This form is useful when you need to graph a function in terms of (r, θ ). You can do this by substituting values of θ into the above equation and then find the corresponding r values.

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