Given the complex number z, start subscript, 1, end subscript, equals, 24, square root of, 2, end square root, left bracket, cosine, start fraction, 23, pi, divided by, 12, end fraction, plus, i, sine, start fraction, 23, pi, divided by, 12, end fraction, right bracketz
1
=24
2
(cos
12
23π
+isin
12
23π
) and z, start subscript, 2, end subscript, equals, 4, left bracket, cosine, start fraction, pi, divided by, 6, end fraction, plus, i, sine, start fraction, pi, divided by, 6, end fraction, right bracketz
2
=4(cos
6
π
+isin
6
π
), express the result of start fraction, z, start subscript, 1, end subscript, divided by, z, start subscript, 2, end subscript, end fraction
z
2
z
1
in rectangular form with fully simplified fractions and radicals.
To find the result of z1/z2, we need to divide z1 by z2:
z1/z2 = (24√2(cos(23π/12) + i sin(23π/12))) / (4(cos(π/6) + i sin(π/6))
To simplify the expression, we multiply the numerator and denominator by the complex conjugate of the denominator:
= [(24√2(cos(23π/12) + i sin(23π/12))) * (4(cos(π/6) - i sin(π/6)))] / [4(cos(π/6) + i sin(π/6)) * (4(cos(π/6) - i sin(π/6))]
= [96√2(cos(23π/12)cos(π/6) + sin(23π/12)sin(π/6)) + 96√2i(sin(23π/12)cos(π/6) - cos(23π/12)sin(π/6))] / [16(cos^2(π/6) + sin^2(π/6))]
= [96√2(cos(29π/12) + i sin(29π/12)) + 96√2i(sin(29π/12) - i cos(29π/12))] / 16
= [96√2cos(29π/12) + 96√2i sin(29π/12) + 96√2i sin(29π/12) + 96√2 cos(29π/12)] / 16
= 192√2 [cos(29π/12) + i sin(29π/12)] / 16
= 12√2 [cos(29π/12) + i sin(29π/12)]
Therefore, z1/z2 = 12√2(cos(29π/12) + i sin(29π/12)) in rectangular form with fully simplified fractions and radicals.