The polar form of a complex number z = a + bi is z = r (cos θ + i sin θ), where r = | z | = √(a2 + b2), a = r cos θ, and b = r sin θ, and θ = tan- 1(b / a) for a > 0 or θ = tan- 1(b / a) + π or θ = tan- 1(b / a) + 180o for a < 0.
\The complex number is z = 3 + 4i.
\The polar form of a complex number z = a + bi is z = r (cos θ + i sin θ).
\Here a = 3 > 0 and b = 4.
\So, first find the absolute value of r .
\r = | z | = √(a2 + b2)
\= √[ (3)2 + (4)2 ]
\= √[ 9 + 16 ]
\= √[ 25 ]
\= 5.
\Now find the argument θ.
\Since a = 3 > 0, use the formula θ = tan- 1(b / a).
\θ = tan- 1(4/3)o
\θ ≅ tan- 1(1.333)o
\θ ≅ 53.13o
\Note that here θ is measured in degrees.
\Therefore, the polar form of 3 + 4i is about 5[ cos(53.13o) + i sin(53.13o) ].
\\