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 $\cos \left( {\dfrac{{n\pi }}{2}} \right) = 0$ , where n= 1, 3, 5, 7, … then ${\cos ^{ - 1}}\left( 0 \right) = \dfrac{\pi }{2}$ .
$\Rightarrow \dfrac{\pi }{2} = 2\theta$
Dividing by 2 on the both sides we have
$\Rightarrow \theta = \dfrac{\pi }{4}$
Therefore, when $r = 0$ we have $\theta = \dfrac{\pi }{4},\dfrac{{3\pi }}{4},\dfrac{{5\pi }}{4},\dfrac{{7\pi }}{4}$
Similarly:
When $\theta = 0$ , we have $r = 0,\dfrac{\pi }{2},\pi ,\dfrac{{3\pi }}{2},2\pi$ .
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.