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Exercises Trigonometry

Example 2 from Algebra and Trigonometry, 10.4 Polar Coordinates: Graphs

Finding Zeros and Maximum Values for a Polar Equation

Using the equation in Example 1, find the zeros and maximum $| r |$ and, if necessary, the polar axis intercepts of $r = 2 \sin θ .$

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
To find the zeros, set $r$ equal to zero and solve for $θ .$
$\begin{array}{l} 2 \sin θ = 0 \\ \sin θ = 0 \\ θ = \sin^{- 1} 0 \\ θ = n π & where n is an integer \end{array}$
Substitute any one of the $θ$ values into the equation. We will use $0.$

$\begin{array}{l} \begin{array}{l} r = 2 \sin (0) \end{array} \\ r = 0 \end{array}$
The points $(0, 0)$ and $(0, \pm n π)$ are the zeros of the equation. They all coincide, so only one point is visible on the graph. This point is also the only polar axis intercept.

To find the maximum value of the equation, look at the maximum value of the trigonometric function $\sin θ,$ which occurs when $θ = \frac{π}{2} \pm 2 k π$ resulting in $r = 3 \cos θ .$ Substitute $\frac{π}{2}$ for $θ.$
$\begin{array}{l} r = 2 \sin (\frac{π}{2}) \\ r = 2 (1) \\ r = 2 \end{array}$

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

Without converting to Cartesian coordinates, test the given equation for symmetry and find the zeros and maximum values of $| r | :$ $r = 3 \cos θ .$

How did it go?

PreviousTesting a Polar Equation for Symmetry Next Sketching the Graph of a Polar Equation for a Circle

From Algebra and Trigonometry (1st edition) by OpenStax (Rice University), section 10.4, 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.