Note that the complex number cos + i sin has absolute value 1 since cos 2 + sin 2 equals 1 for any angle .Thus, every complex number z is the product of a real number |z| and a complex number cos + i sin .. We’re almost to the point where we can prove the last unproved statement of the previous section on multiplication, namely, that arg(zw) = arg(z) + arg(w). Convert r and theta back into the original complex number. The formula for converting rectangular coordinates to radius , follows immediately from the Pythagorean theorem, while the follows from the definition of the tangent function itself. If we let rbe the distance of zfrom the origin and, if z6=0 ,we let θbe the angle that the line connecting zto the origin makes with the positive real axis then we can write z= x+iy= rcosθ+irsinθ. Answer. Hence, a r g a r c t a n () = − √ 3 + = − 3 + = 2 3. 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