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Writing a cartesian equation in polar form

Write the Cartesian equation x 2 + y 2 = 9 in polar form.

The goal is to eliminate x and y from the equation and introduce r and θ . Ideally, we would write the equation r as a function of θ . To obtain the polar form, we will use the relationships between ( x , y ) and ( r , θ ) . Since x = r cos θ and y = r sin θ , we can substitute and solve for r .

    ( r cos θ ) 2 + ( r sin θ ) 2 = 9     r 2 cos 2 θ + r 2 sin 2 θ = 9      r 2 ( cos 2 θ + sin 2 θ ) = 9                            r 2 ( 1 ) = 9   Substitute cos 2 θ + sin 2 θ = 1.                                   r = ± 3 Use the square root property .

Thus, x 2 + y 2 = 9 , r = 3 , and r = 3 should generate the same graph. See [link] .

Plotting a circle of radius 3 with center at the origin in polar and rectangular coordinates. It is the same in both systems.
(a) Cartesian form x 2 + y 2 = 9 (b) Polar form r = 3

To graph a circle in rectangular form, we must first solve for y .

x 2 + y 2 = 9          y 2 = 9 x 2            y = ± 9 x 2

Note that this is two separate functions, since a circle fails the vertical line test. Therefore, we need to enter the positive and negative square roots into the calculator separately, as two equations in the form Y 1 = 9 x 2 and Y 2 = 9 x 2 . Press GRAPH.

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Rewriting a cartesian equation as a polar equation

Rewrite the Cartesian equation x 2 + y 2 = 6 y as a polar equation.

This equation appears similar to the previous example, but it requires different steps to convert the equation.

We can still follow the same procedures we have already learned and make the following substitutions:

r 2 = 6 y Use  x 2 + y 2 = r 2 . r 2 = 6 r sin θ Substitute y = r sin θ .         r 2 6 r sin θ = 0 Set equal to 0 .        r ( r 6 sin θ ) = 0 Factor and solve . r = 0 We reject  r = 0 , as it only represents one point,  ( 0 , 0 ) . or r = 6 sin θ

Therefore, the equations x 2 + y 2 = 6 y and r = 6 sin θ should give us the same graph. See [link] .

Plots of the equations stated above - the plots are the same in both rectangular and polar coordinates. They are circles.
(a) Cartesian form x 2 + y 2 = 6 y (b) polar form r = 6 sin θ

The Cartesian or rectangular equation is plotted on the rectangular grid, and the polar equation is plotted on the polar grid. Clearly, the graphs are identical.

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Rewriting a cartesian equation in polar form

Rewrite the Cartesian equation y = 3 x + 2 as a polar equation.

We will use the relationships x = r cos θ and y = r sin θ .

                         y = 3 x + 2                   r sin θ = 3 r cos θ + 2 r sin θ 3 r cos θ = 2 r ( sin θ 3 cos θ ) = 2 Isolate  r .                        r = 2 sin θ 3 cos θ Solve for  r .
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Rewrite the Cartesian equation y 2 = 3 x 2 in polar form.

r = 3

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Identify and graph polar equations by converting to rectangular equations

We have learned how to convert rectangular coordinates to polar coordinates, and we have seen that the points are indeed the same. We have also transformed polar equations to rectangular equations and vice versa. Now we will demonstrate that their graphs, while drawn on different grids, are identical.

Graphing a polar equation by converting to a rectangular equation

Covert the polar equation r = 2 sec θ to a rectangular equation, and draw its corresponding graph.

The conversion is

            r = 2 sec θ          r = 2 cos θ   r cos θ = 2 x = 2

Notice that the equation r = 2 sec θ drawn on the polar grid is clearly the same as the vertical line x = 2 drawn on the rectangular grid (see [link] ). Just as x = c is the standard form for a vertical line in rectangular form, r = c sec θ is the standard form for a vertical line in polar form.

Plots of the equations stated above - the plots are the same in both rectangular and polar coordinates. They are lines.
(a) Polar grid (b) Rectangular coordinate system

A similar discussion would demonstrate that the graph of the function r = 2 csc θ will be the horizontal line y = 2. In fact, r = c csc θ is the standard form for a horizontal line in polar form, corresponding to the rectangular form y = c .

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Questions & Answers

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
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2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
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can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
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Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
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Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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