Question

Convert r=4cos(teta) to rectangular form?

Answer 1

Use \(x=r\, \cos (\theta ) \Leftrightarrow \cos (\theta )=\frac{x}{r}\).

Answer 2

\[r=4\cos (\theta )=4\frac{x}{r}\]\[r^2=4x\]Does that make sense so far?

Answer 3

Then use \(r^2=x^2+y^2\) (this is exactly the pythagorean theorem, btw).

Answer 4

\[x^2+y^2=4x\]\[y^2=4x-x^2\]\[\large y =\pm \sqrt{4x-x^2}\]

Answer 5

You can continue from here, as well. The expression under the radical is of the form: \(-(x^2+bx)\). You may desire to complete the square:\[x^2-4x=(x^2-4x+4)-4=(x-2)^2-4\]\[4x-x^2=-(x^2-4x)=4-(x-2)^2\]\[\large y=\pm \sqrt{4-(x-2)^2}\]Which is the rectangular equation for a circle of radius 2 centered at (2,0).