Answer:
Correct option (D) x = 0 or x² + (y - 1)² = 1².
Step-by-step explanation:
The trigonometric equation is: [tex]r\ Cos\ \theta=Sin\ 2\theta[/tex]
The expansion of [tex]Sin\ 2\theta[/tex] is:
[tex]Sin\ 2\theta=2\ Sin\ \theta \ Cos\ \theta[/tex]
Then the equation is:
[tex]r\ Cos\ \theta=Sin\ 2\theta\\r\ Cos\ \theta=2\ Sin\ \theta \ Cos\ \theta[/tex]
Now either [tex]Cos\ \theta=0[/tex] or [tex]r=2Sin\ \theta[/tex]
In Cartesian form:
[tex]x=Cos\ \theta\\y=Sin\ \theta[/tex]
Then either
x = 0
Or,
[tex]\sqrt{x^{2}+y^{2}}=\frac{2y}{\sqrt{x^{2}+y^{2}}}\\x^{2}+y^{2}-2y=0\\x^{2}+y^{2}-2y+1=1\\x^{2}+(y-1)^{2}=1[/tex]
Thus, the resulting curve is: x = 0 or x² + (y - 1)² = 1².
