Question

How do I graph the function of $$r=\sin 3\theta$$ ?

Answer 1

Hint: Here in this question, we have to plot the graph of the given trigonometric equation. To plot the graph first we have to find the coordinate $$\left(r,\theta \right)$$ by comparing the general equation of the rose curve i.e., $$r=a\sin 3\theta$$ . By finding the coordinate we can plot the required graph of given trigonometric equation

Complete step-by-step answer:
In general let us consider $$r=a\sin (n\theta )$$ or $$r=a\sin (n\theta )$$ where $$a\ne 0$$ and n is a positive number greater than 1. For the graph of rose if the value of n is odd then rose will have n petals or if the value of n is even then the rose will have 2n petals. Here “a” represents the radius of the circle where the rose petals lie.
Now consider the given equation $$r=\sin 3\theta$$ . Here a=1, the radius of the circle is 1 and n=3, the number is odd so we have 3 petals for the rose.
Now consider the given equation $$r=\sin (3\theta )$$ ------- (1)
Substitute r=0 in equation (1) we have
 $$\Rightarrow 0=\sin (3\theta )$$
By taking the inverse we have
 $$\Rightarrow {\sin }^{-1}(0)=3\theta$$
By the table of trigonometry ratios for standard angles in radians we have $$\sin \left(n\pi \right)=0$$ , but here n = 1 so we have
 $$\Rightarrow \pi =3\theta$$
Dividing by 3 on the both sides we have
 $$\Rightarrow \theta ={\displaystyle \frac{\pi }{3}}$$
Therefore $$\left(\alpha ,\beta \right)=\left(r,\theta \right)=\left(0,{\displaystyle \frac{\pi }{3}}\right)$$
While determining the area we use the above coordinates
Hence the graph of the given rose curve equation $$r=\sin 3\theta$$ is:

Note: Here we have to plot the polar graph. The polar graph is plotted versus $$r$$ and $$\theta$$ . By substituting the value of $$\theta$$ we can determine the value of $$r$$ . Here a=1, the radius of the circle is 1 and n=3, the number is odd so we have 3 petals for the rose. The petals will not exceed the circle of radius