How do you convert (3, - 4) (3,−4) into polar coordinates?

1 Answer
Alan P.
Jan 4, 2016

(r,theta)=(5,2pi+arctan(-3/4))(r,θ)=(5,2π+arctan(−34)) (approximately =(5,5.64)=(5,5.64)

Explanation:

The radius rr is given by the Pythagorean Theorem as
color(white)("XXX")r=sqrt((-4)^2+3^2) = 5XXXr=√(−4)2+32=5

The value of thetaθ is based on the arctan(y/x)arctan(yx) but is dependent upon the quadrant in which (x,y)(x,y) is positioned. This is because the arctanarctan function returns a value in the range [-pi/2,+pi/2][−π2,+π2] relative to the X-axis.

{: (color(black)("Quadrant"),,color(white)("XXX")color(black)(theta)), (bar(color(white)("XXXXXX")),,bar(color(white)("XXXXXX"))), (I,,arctan(y/x)), (II,,arctan(y/x)+pi), (III,,arctan(y/x)+pi), (IV,,arctan(y/x)+2pi) :}

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