I'm not sure if I'm doing this right so far I found r=0 theta=0 r=5 theta 45 r=0 theta 90 so on up to 675. How do I plot this and find the tips on an 8 pedal rose?

Not quite right. Since \sin \alpha = 0 for \alpha = 0,\ \pi,\ \2\pi, etc. then for the equation R = 5 \sin 4 \theta , R=0 when:


\theta = 0, \\
\theta =\frac {\pi} 4 , \\
\theta =\frac {\pi} 2, \\
\theta = \frac {3\pi} 4

etc.

And R = +5 for: \theta = \frac {\pi}8, \ \frac {5 \pi} 8 , etc. and -5 for: \theta = \frac {3 \pi} 8, \ \frac {7 \pi} 8,etc.

Can you plot this out now?

Why is it pi/8 5pi/8... how did you formulate these angles?

If R = 5, you get:

5\sin4\theta = 5

\sin4\theta = 1

\alpha = \sin^{-1}(1)

And \alpha = \frac{\pi}{2}, \frac{3\pi}{2},\frac{5\pi}{2}, \frac{7\pi}{2},\frac{9\pi}{2}, \frac{11\pi}{2},\frac{13\pi}{2}, \frac{15\pi}{2}

So,

4\theta = \frac{\pi}{2}, \frac{3\pi}{2},\frac{5\pi}{2}, \frac{7\pi}{2},\frac{9\pi}{2}, \frac{11\pi}{2},\frac{13\pi}{2}, \frac{15\pi}{2}

\theta = \frac{\pi}{8}, \frac{3\pi}{8},\frac{5\pi}{8}, \frac{7\pi}{8},\frac{9\pi}{8}, \frac{11\pi}{8},\frac{13\pi}{8}, \frac{15\pi}{8}

Since \small \sin \frac {\pi} 2 \pm 2 \pi n=1 , where n = 0, 1, 2, 3, etc. then \small \sin 4 \theta = 1 for \small \theta = \frac {\pi} 8 \pm \frac {\pi n} 2. Hence \small \theta = \frac {\pi} 8 , \frac {5 \pi} 8, \frac {9 \pi} 8 ,, etc.

Similarly, \small \sin \frac {3 \pi} 2 \pm 2 \pi n= -1, and so \small \sin 4 \theta = -1 for \small \theta = \frac {3\pi} 8 \pm \frac {\pi } 2. Hence \small \theta = \frac {3 \pi} 8 , \frac {7 \pi} 8, \frac {11 \pi} 8 ,, etc.

To draw your 8-pointed pedal, note that the radius for various angles are:


\begin {matrix}
Angle & Radius \\
\\
0 & 0 \\
\\
\frac {\pi} 8 & 5 \\
\\
\frac {\pi} 4 & 0 \\
\\
\frac {3 \pi} 8 &-5 \\
\\
\frac {\pi} 2 & 0 \\
\\
\frac {5 \pi} 8 & 5 \\
\\
\frac {3 \pi} 4 & 0 \\
\\
\frac {7 \pi} 8 & -5
\end{matrix}