Question

Consider the numbers z = 3 startroot 2 endroot(cos(135°) i sin(135o) and w = cos(180o) i sin(180°). which expression is closest to the polar form of z w? 5(cos(143o) i sin(143o)) 5(cos(315o) i sin(315o)) 4startroot 2 endroot(cos(143o) i sin(143o)) 4startroot 2 endroot(cos(315o) i sin(315o))

Answer 1

The polar form of the product of complex numbers z and w is calculated by multiplying their magnitudes and summing their angles. The correct expression closest to the polar form of z multiply w is 4√2(cos(315°) + i sin(315°)).

Step-by-step explanation:

To find the polar form of the product of two complex numbers, you simply multiply their magnitudes and add their angles. The question provides the complex numbers in polar form:

z = 3√2(cos(135°) + i sin(135°))

w = cos(180°) + i sin(180°)

The magnitude of w is 1 (since cos(180°) = -1 and sin(180°) = 0), so we can ignore it when multiplying magnitudes. To find the angle for w, recall that cos(180°) = -1 falls on the negative x-axis, and sin(180°) = 0 falls on the x-axis, thus the angle is 180°. Therefore, when we multiply z and w, the resulting angle is the sum of the angles from z and w, which is 135° + 180° = 315°. The magnitude of the product is just the product of the magnitudes, so it's 3√2 × 1 = 3√2.

Therefore, the polar form of z×w is 3√2(cos(315°) + i sin(315°)). Among the given options, 4√2(cos(315°) + i sin(315°)) is closest to our result.