Question

Write each of the given numbers in the polar form reiθ,−π<θ≤π.

(a) (cos9−2π​+isin9−2π​)3 r=
(b) −3​+i6+6i​ r=,θ=
(c) 7e(8+i)4i​
Write the following numbers in the polar form reiθ,0≤θ<2π :
(a) 51​
(b) 7= θ=
(c) −0+6i θ=
Note: You can eam partial credit on this problem.

Answer 1

To write numbers in polar form, we need to find their magnitude and angle. By applying the necessary formulas, we can find the polar form of the given complex numbers. For the numbers in (a), (b), and (c), the polar forms are 1*e^(i3(9-2π)), sqrt(117)*e^(-iπ/4*3), and 7*e^(i(8-i)) respectively. In (a), (b), and (c), the polar forms of 51, 7, and -0+6i are 51*e^(i0), 7*e^(i0), and 6*e^(i(π/2)).

Step-by-step explanation:

Given (a) (cos(9-2π) + i*sin(9-2π))^3, we can express the number in polar form as r*e^(iθ), where r is the magnitude of the number and θ is the angle it makes with the positive real axis. In this case, r = |cos(9-2π) + i*sin(9-2π)| = 1 and θ = 9-2π. Therefore, the polar form of (cos(9-2π) + i*sin(9-2π))^3 is 1*e^(i(9-2π)*3) = e^(i3(9-2π)).

In (b) (-3+i6+6i)^3, we can find the magnitude r = |-3+i6+6i| = sqrt((-3)^2+(6)^2+(6)^2) = sqrt(117) and the angle θ = arg(-3+i6+6i) = arctan((6+6)/(-3)) = -π/4. Therefore, the polar form of (-3+i6+6i)^3 is sqrt(117)*e^(-iπ/4*3).

In (c) 7e^(8+i)^4i, we can find the magnitude r = |7e^(8+i)^4i| = 7 and the angle θ = arg(7e^(8+i)^4i) = 8+i - 4i = 8-i. Therefore, the polar form of 7e^(8+i)^4i is 7*e^(i(8-i)).

In (a) 51, we can express the number in polar form by finding r = |51| = 51 and θ = arg(51) = 0. Therefore, the polar form of 51 is 51*e^(i(0)).

In (b) 7, we can express the number in polar form by finding r = |7| = 7 and θ = arg(7) = 0. Therefore, the polar form of 7 is 7*e^(i(0)).

In (c) -0+6i, we can express the number in polar form by finding r = |-0+6i| = 6 and θ = arg(-0+6i) = π/2. Therefore, the polar form of -0+6i is 6*e^(i(π/2)).

Answer 2

The provided problem seems to require converting complex numbers to their polar form, which includes a modulus and an angle, but the actual complex numbers to convert were not presented clearly due to possible typos or formatting errors.

Step-by-step explanation:

The question seems to involve converting complex numbers into their polar form, reiθ, where r represents the magnitude of the complex number and θ represents the argument or angle in radians. Calculating r involves finding the modulus of the complex number, and θ is the angle made with the positive x-axis in the Argand diagram.

Polar form is useful for complex number arithmetic, particularly for multiplication and division. However, the question as presented seems to have typos or formatting issues, so it's unclear which specific complex numbers need converting to polar form.