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Exercises Trigonometry

Example 10 from Algebra and Trigonometry, 10.4 Polar Coordinates: Graphs

Sketching the Graph of an Archimedes’ Spiral

Sketch the graph of $r = θ$ over $[0, 2 π] .$

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
As $r$ is equal to $θ,$ the plot of the Archimedes’ spiral begins at the pole at the point (0, 0). While the graph hints of symmetry, there is no formal symmetry with regard to passing the symmetry tests. Further, there is no maximum value, unless the domain is restricted.
Create a table such as Table 10.

$θ$ $\frac{π}{4}$ $\frac{π}{2}$ $π$ $\frac{3 π}{2}$ $\frac{7 π}{4}$ $2 π$

$r$ 0.785 1.57 3.14 4.71 5.50 6.28

Table 10

Notice that the r-values are just the decimal form of the angle measured in radians. We can see them on a graph in Figure 19.

Figure 19 Archimedes’ spiral

Your turn

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

Sketch the graph of $r = - θ$ over the interval $[0, 4 π] .$

How did it go?

PreviousSketching the Graph of a Rose Curve (n Odd)Next Plotting a Complex Number in the Complex Plane

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.