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Exercises Trigonometry

Example 9 from Algebra and Trigonometry, 10.3 Polar Coordinates

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.

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
The conversion is

$\begin{matrix} r & = & 2 \sec θ \\ r & = & \frac{2}{\cos θ} \\ r \cos θ & = & 2 \\ x & = & 2 \end{matrix}$
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 Figure 12). 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.

Figure 12 (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 $y = c .$

How did it go?

PreviousRewriting a Cartesian Equation in Polar Form Next Rewriting a Polar Equation in Cartesian 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.