Question

How do I graph the function of $$r=\cos 2\theta$$ ?

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., $$r=a\cos \left(n\theta \right)$$ . By finding the coordinate we can plot the required graph of given trigonometric equation

Complete step by step solution:
In general let we consider $$r=a\cos \left(n\theta \right)$$ 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 lies.
Now consider the given equation $$r=\cos 2\theta$$ . Here a=1, the radius of circle is 1 and n=2, the number is even so we have 2n petals i.e., 4 petals for the rose.
Now consider the given equations ------- (1)
Substitute r=0 in equation (1) we have
 $$\Rightarrow 0=\cos 2\theta$$
By taking the inverse we have
 $$\Rightarrow {\cos }^{-1}(0)=2\theta$$
By the table of trigonometry ratios for standard angles in radians we have $$\cos \left({\displaystyle \frac{n\pi }{2}}\right)=0$$ , where n= 1, 3, 5, 7, … then $${\cos }^{-1}\left(0\right)={\displaystyle \frac{\pi }{2}}$$ .
 $$\Rightarrow {\displaystyle \frac{\pi }{2}}=2\theta$$
Dividing by 2 on the both sides we have
 $$\Rightarrow \theta ={\displaystyle \frac{\pi }{4}}$$
Therefore, when $$r=0$$ we have $$\theta ={\displaystyle \frac{\pi }{4}},{\displaystyle \frac{3\pi }{4}},{\displaystyle \frac{5\pi }{4}},{\displaystyle \frac{7\pi }{4}}$$
Similarly:
When $$\theta =0$$ , we have $$r=0,{\displaystyle \frac{\pi }{2}},\pi ,{\displaystyle \frac{3\pi }{2}},2\pi$$ .
While determining the area we use the above coordinates
Hence the graph of the given rose curve equation $$r=\cos 2\theta$$ 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.