# How do you graph r=10sin4theta?

Aug 10, 2018

See explanation and graph.

#### Explanation:

$0 \le r = 10 \sin 4 \theta \in \left[0 , 10\right]$ and the period = $\frac{2 \pi}{4} = \frac{\pi}{2}$.

As r is non-negative, so is $\sin 4 \theta$, and so,

$4 \theta \notin \left[\pi , 2 \pi\right] \Rightarrow \theta \notin \left[\frac{\pi}{4} , \frac{\pi}{2}\right] \Rightarrow r \ge 0$ for

only half of every period.

Four loops are created for

$\theta \in \left[0 , \frac{\pi}{4}\right] , \left[\frac{\pi}{2} , \frac{3}{4} \pi\right] . \left[\pi , \frac{5}{4} \pi\right] \mathmr{and} \left[\frac{3}{2} \pi , \frac{7}{4} \pi\right]$.

See graph depicting these aspects now, for the converted

equation, using

$\sin 4 \theta = 4 \left({\cos}^{3} \theta \sin \theta - \cos \theta {\sin}^{3} \theta\right)$

${\left({x}^{2} + {y}^{2}\right)}^{2.5} = 10 \left(4 \left({x}^{3} y - x {y}^{3}\right)\right)$
graph{( x^2 + y^2 )^2.5 - 40 (x^3y - xy^3 ) = 0 [ -20 20 -10 10]}

I think, after reading again and again that scalar $r \ge 0$ in my

answers, the centuries old practice of showing ( 4 r-positive + 4 r-

negative ) 8 loops, for this equation, is given a go.