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Exercises Trigonometry

Example 7 from Algebra and Trigonometry, 10.3 Polar Coordinates

Rewriting a Cartesian Equation as a Polar Equation

Rewrite the Cartesian equation $x^{2} + y^{2} = 6 y$ as a polar equation.

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Solution
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:

$\begin{array}{l} r^{2} = 6 y & Use x^{2} + y^{2} = r^{2} . \\ r^{2} = 6 r \sin θ & Substitute y = r \sin θ . \\ r^{2} - 6 r \sin θ = 0 & Set equal to 0 . \\ r (r - 6 \sin θ) = 0 & Factor and solve . \\ r = 0 & We reject r = 0, as it only represents one point, (0, 0) . \\ r = 6 \sin θ \begin{matrix} \end{matrix} \end{array}$
Therefore, the equations $x^{2} + y^{2} = 6 y$ and $r = 6 \sin θ$ should give us the same graph. See Figure 11.

Figure 11 (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.

How did it go?

PreviousWriting a Cartesian Equation in Polar Form Next Rewriting a Cartesian Equation in Polar Form

From Algebra and Trigonometry (1st edition) by OpenStax (Rice University), section 10.3, licensed under CC BY 4.0. Original at openstax.org. We split the solution into steps; the words, numbers and figures are the book's. OpenStax doesn't endorse this site.