How do you graph $r = - 8 sin 3 θ$ $r=-8sin3theta$ ?

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Answer 1 · 2016-12-27T21:58:32

It has a 3 blade propeller shape.

I tried graphing $r = - 8 sin (3 θ)$ $r = -8sin(3theta)$ in both polar and Cartesian coordinates; neither is easy.

I used WolframAlpha to convert to functions of $θ$ $theta$ :

$r = 8 {sin}^{3} (θ) - 24 sin (θ) {cos}^{2} (θ)$ $r = 8sin^3(theta) - 24sin(theta)cos^2(theta)$

Multiply both sides by $r^{3}$ $r^3$ :

$r^{4} = 8 r^{3} {sin}^{3} (θ) - 24 r sin (θ) r^{2} {cos}^{2} (θ)$ $r^4 = 8r^3sin^3(theta) - 24rsin(theta)r^2cos^2(theta)$

Applying the polar to rectangular conversion equations:

${(x^{2} + y^{2})}^{2} = 8 y^{2} - 24 y (x^{2})$ $(x^2 + y^2)^2 = 8y^2 - 24y(x^2)$

Here is the graph:

![Desmos.com](useruploads.socratic.org)

How do you graph r=−8sin3θr=-8sin3theta?