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Exercises Trigonometry

Example 5 from Algebra and Trigonometry, 10.3 Polar Coordinates

Writing Rectangular Coordinates as Polar Coordinates

Convert the rectangular coordinates $(3, 3)$ to polar coordinates.

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
We see that the original point $(3, 3)$ is in the first quadrant. To find $θ,$ use the formula $\tan θ = \frac{y}{x} .$ This gives

$\begin{array}{l} \tan θ = \frac{3}{3} \\ \tan θ = 1 \\ θ = \tan^{−1} (1) \\ θ = \frac{π}{4} \end{array}$
To find $r,$ we substitute the values for $x$ and $y$ into the formula $r = \sqrt{x^{2} + y^{2}} .$ We know that $r$ must be positive, as $\frac{π}{4}$ is in the first quadrant. Thus

$\begin{array}{l} r = \sqrt{3^{2} + 3^{2}} \\ r = \sqrt{9 + 9} \\ r = \sqrt{18} = 3 \sqrt{2} \end{array}$
So, $r = 3 \sqrt{2}$ and $θ = \frac{π}{4},$ giving us the polar point $(3 \sqrt{2}, \frac{π}{4}) .$ See Figure 9.

Figure 9

How did it go?

PreviousWriting Polar Coordinates as Rectangular Coordinates Next Writing 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.