How do you graph $r=1+2\sin \theta $ ?

Question

Vedantu Content Team · Accepted Answer

Hint: For getting the graph of the given question, we will have to apply the value of one variable for getting the value of another variable. Since, in the question, there are two variables $r$ and $\theta $ . So, we will put the value for $\theta $ and after solving the equation, we will get the value of $r$ . Then, after using these values in the graph that will be the representative of the equation.Complete step by step solution:Since, the given equation that will help us to get the graph as:$\Rightarrow r=1+2\sin \theta $Since, $\theta $ represents the angle. So, we will put the value of that angle that will help us to get the value of $\sin \theta $ easily.Here, we will put the value for $\theta $ so that we can get the value of $r$ as:Let us put $\theta =0{}^\circ $ in the equation as:$\Rightarrow r=1+2\sin 0{}^\circ $$\Rightarrow r=1+2\times 0$$\Rightarrow r=1+0$$\Rightarrow r=1$Now, let us  $\theta =30{}^\circ $ in the question as:$\Rightarrow r=1+2\sin 30{}^\circ $$\Rightarrow r=1+2\times \dfrac{1}{2}$$\Rightarrow r=1+1$$\Rightarrow r=2$Here, let the value of $\theta =45{}^\circ $$\Rightarrow r=1+2\sin 45{}^\circ $$\Rightarrow r=1+2\times \dfrac{1}{\sqrt{2}}$$\Rightarrow r=1+\sqrt{2}$$\Rightarrow r=1+1.4$$\Rightarrow r=2.4$Now, let the value for $\theta =60{}^\circ $$\Rightarrow r=1+2\sin 60{}^\circ $$\Rightarrow r=1+2\times \dfrac{\sqrt{3}}{2}$$\Rightarrow r=1+\sqrt{3}$$\Rightarrow r=1+1.7$$\Rightarrow r=2.7$Now, assume the value of variable $\theta =90{}^\circ $$\Rightarrow r=1+2\sin 90{}^\circ $$\Rightarrow r=1+2\times 1$$\Rightarrow r=1+2$$\Rightarrow r=3$Here, we will make the table for these variables so that we can easily get the graph as:Value for $\theta $Value of $r$$0{}^\circ $$1$$30{}^\circ $2$45{}^\circ $2.4$60{}^\circ $2.7$90{}^\circ $3Now, we got some values for $\theta $ and $r$ . So, the graphical representation will be as:

Note: Here, we will check if the solution of the equation is correct or not by putting the obtained values of $r$ and evaluating the values of $\theta $ as:Value of $r$Equation:$\Rightarrow r=1+2\sin \theta $Value for $\theta $$1$$\Rightarrow 1=1+2\sin \theta $$\Rightarrow 2\sin \theta =1-1$$\Rightarrow 2\sin \theta =0$$\Rightarrow \sin \theta =0$$\Rightarrow \sin \theta =\sin 0{}^\circ $$\Rightarrow \theta =0{}^\circ $$0{}^\circ $2$\Rightarrow 2=1+2\sin \theta $$\Rightarrow 2\sin \theta =2-1$$\Rightarrow 2\sin \theta =1$$\Rightarrow \sin \theta =\dfrac{1}{2}$$\Rightarrow \sin \theta =\sin 30{}^\circ $$\Rightarrow \theta =30{}^\circ $$30{}^\circ $2.4$\Rightarrow 2.4=1+2\sin \theta $$\Rightarrow 2\sin \theta =2.4-1$$\Rightarrow 2\sin \theta =1.4$$\Rightarrow 2\sin \theta =\sqrt{2}$$\Rightarrow \sin \theta =\dfrac{\sqrt{2}}{2}$$\Rightarrow \sin \theta =\dfrac{1}{\sqrt{2}}$$\Rightarrow \sin \theta =\sin 45{}^\circ $$\Rightarrow \theta =45{}^\circ $$45{}^\circ $2.7$\Rightarrow 2.7=1+2\sin \theta $$\Rightarrow 2\sin \theta =2.7-1$$\Rightarrow 2\sin \theta =1.7$$\Rightarrow 2\sin \theta =\sqrt{3}$$\Rightarrow \sin \theta =\dfrac{\sqrt{3}}{2}$$\Rightarrow \sin \theta =\sin 60{}^\circ $$\Rightarrow \theta =60{}^\circ $$60{}^\circ $3$\Rightarrow 3=1+2\sin \theta $$\Rightarrow 2\sin \theta =3-1$$\Rightarrow 2\sin \theta =2$$\Rightarrow \sin \theta =\dfrac{2}{2}$$\Rightarrow \sin \theta =1$$\Rightarrow \sin \theta =\sin 90{}^\circ $$\Rightarrow \theta =90{}^\circ $$90{}^\circ $.

Value for $\theta $	Value of $r$
$0{}^\circ $	$1$
$30{}^\circ $	2
$45{}^\circ $	2.4
$60{}^\circ $	2.7
$90{}^\circ $	3

Value of $r$	Equation:$\Rightarrow r=1+2\sin \theta $	Value for $\theta $
$1$	$\Rightarrow 1=1+2\sin \theta $$\Rightarrow 2\sin \theta =1-1$$\Rightarrow 2\sin \theta =0$$\Rightarrow \sin \theta =0$$\Rightarrow \sin \theta =\sin 0{}^\circ $$\Rightarrow \theta =0{}^\circ $	$0{}^\circ $
2	$\Rightarrow 2=1+2\sin \theta $$\Rightarrow 2\sin \theta =2-1$$\Rightarrow 2\sin \theta =1$$\Rightarrow \sin \theta =\dfrac{1}{2}$$\Rightarrow \sin \theta =\sin 30{}^\circ $$\Rightarrow \theta =30{}^\circ $	$30{}^\circ $
2.4	$\Rightarrow 2.4=1+2\sin \theta $$\Rightarrow 2\sin \theta =2.4-1$$\Rightarrow 2\sin \theta =1.4$$\Rightarrow 2\sin \theta =\sqrt{2}$$\Rightarrow \sin \theta =\dfrac{\sqrt{2}}{2}$$\Rightarrow \sin \theta =\dfrac{1}{\sqrt{2}}$$\Rightarrow \sin \theta =\sin 45{}^\circ $$\Rightarrow \theta =45{}^\circ $	$45{}^\circ $
2.7	$\Rightarrow 2.7=1+2\sin \theta $$\Rightarrow 2\sin \theta =2.7-1$$\Rightarrow 2\sin \theta =1.7$$\Rightarrow 2\sin \theta =\sqrt{3}$$\Rightarrow \sin \theta =\dfrac{\sqrt{3}}{2}$$\Rightarrow \sin \theta =\sin 60{}^\circ $$\Rightarrow \theta =60{}^\circ $	$60{}^\circ $
3	$\Rightarrow 3=1+2\sin \theta $$\Rightarrow 2\sin \theta =3-1$$\Rightarrow 2\sin \theta =2$$\Rightarrow \sin \theta =\dfrac{2}{2}$$\Rightarrow \sin \theta =1$$\Rightarrow \sin \theta =\sin 90{}^\circ $$\Rightarrow \theta =90{}^\circ $	$90{}^\circ $.