Question

When a complex number is in trigonometric form , 
what does  represent? 

	A.	abscissa
	B.	argument
	C.	imaginary unit
	D.	modulus

Find  and write the answer in standard form. 

	A.
	B.
	C.
	D.

Find  and write the answer in standard form. 

	A.
	B.
	C.
	D.

 

Answer 1

1)

The complex number

z = r(cos θ + i sin θ)

θ represents argument of the complex number .

Option B is correct.

Answer 2

 [ 5 (cos20° + i sin20°)]³

Using De Moivre’s theorem:

If z =  r(cos θ + i sin θ) then zⁿ = [r(cos θ + i sin θ)]ⁿ =[rⁿ(cos nθ + i sin nθ)]

[ 5 (cos20° + i sin20°)]³ = 5³(cos 3(20°) + i sin3(20°))

                                               =125(cos 60° + i sin60°)

                                               =125[(1/2) + i (√3/2)]

                                                =125[(1/2) + i (√3/2)]

                                     =(125/2) + i 125(√3/2)

Option C is correct.

Answer 3

2 (√3 + i)⁷

Convert the √3 + i to polar form:

Find the magnitude of the complex number.

r = √[(√3)² +(1)²]

   = √[3 +1]

   = 2

Find the argument of the complex number.

θ = tan^-1 (1/√3) = 30°

√3 + i = 2 (cos30° + i sin30°)

Using De Moivre’s theorem:

If z =  r(cos θ + i sin θ) then zⁿ = [r(cos θ + i sin θ)]ⁿ =[rⁿ(cos nθ + i sin nθ)]

2 (√3 + i)⁷ = 2[2 (cos30° + i sin30°)]⁷

                     = 2(2)⁷(cos 7(30°) + i sin7(30°))

                =(2)⁸(cos 210° + i sin210°)

                 =(2)⁸(cos (180°+30°) + i sin(180°+30°))

                      =(2)⁸(- cos 30° - i sin 30°)

                      =256(- (√3/2) - i (1/2))

                 =-128√3 -128 i

Option D is correct.