Question

How do you graph $ r=2.\sin \theta $ ?

Answer 1

Hint: $ \left( r,\theta \right) $ is polar coordinate form. If (x,y) is Cartesian coordinate form then r is the distance of the point from the origin, and $ \theta $ is the angle between the line joining origin and the point and positive x-axis. We can write r and $ \theta $ in terms of x and y then write the function and draw the graph.

Complete step by step answer:
The function is given in the polar form. Let’s understand what polar form is.
Let’s draw a Cartesian plane with coordinates (x,y).

(x,y) is the cartesian coordinate form.
The polar coordinate $ \left( r,\theta \right) $ . r is the distance between origin and (x,y). $ \theta $ is the angle between
The line joining origin and (x,y) and positive x-axis.
We can see the value of r is equal to $ \sqrt{{{x}^{2}}+{{y}^{2}}} $ the value of $ \theta $ is $ {{\sin }^{-1}}\dfrac{y}{\sqrt{{{x}^{2}}+{{y}^{2}}}} $
If we replace r as $ \sqrt{{{x}^{2}}+{{y}^{2}}} $ and $ \sin \theta $ as $ \dfrac{y}{\sqrt{{{x}^{2}}+{{y}^{2}}}} $ in the equation $ r=2.\sin \theta $ we will get
 $ \sqrt{{{x}^{2}}+{{y}^{2}}}=\dfrac{2y}{\sqrt{{{x}^{2}}+{{y}^{2}}}} $
Further solving we get
 $ \Rightarrow {{x}^{2}}+{{y}^{2}}=2y $
 $ \Rightarrow {{x}^{2}}+{{y}^{2}}-2y=0 $
 $ \Rightarrow {{x}^{2}}+{{y}^{2}}-2y+1=1 $
 $ \Rightarrow {{\left( x-0 \right)}^{2}}+{{\left( y-1 \right)}^{2}}=1 $
Now we can see the above equation is the equation of circle with center at (0,1) and radius is 1 unit.
Now we can draw the graph as below,

In the above figure, the center of the circle is shown as O (0,1) radius is 1. So this one way we can draw the graph of the polar coordinate equation.

Note:
Another way of drawing the graph is manual way first we have note some $ \left( r,\theta \right) $ then mark the corresponding point and join all the points for example in this case we choose (0,0) , $ \left( 1,\dfrac{\pi }{6} \right),\left( \sqrt{3},\dfrac{\pi }{3} \right),\left( 2,\dfrac{\pi }{2} \right) $ we have to choose a point at distance 1 unit in angle $ \dfrac{\pi }{6} $ , similarly all other points and join them.