Question

What is the result of multiplication of complex numbers - √3+ i and
1/8 (cos 5π/3+ i sin 5π/3)?​

Answer 1

We have to find the the result of multiplication of complex number -√3 + i and 1/8(cos5π/3 + i sin5π/3)

solution : at first, convert 1/8(cos5π/3 + i sin5π/3) into a + ib form.

cos(5π/3) = cos(2π - π/3) = cos(π/3) = 1/2

sin(5π/3) = sin(2π - π/3) = -sin(π/3) = -√3/2

so, 1/8(cos5π/3 + i sin5π/3) = 1/8(1/2 + i (-√3/2))

= 1/16 - √3/16 i

now multiplication of -√3 + i and 1/16 - √3/16 i

= (-√3 + i) × (1/16 - √3/16 i)

= -√3/16 + 3/16i + 1/16 i - √3/16 i²

= -√3/16 + 4/16 i - √3/16 (-1) [ ∵ i² = -1 ]

= 1/4 i

Therefore the result of multiplication of given complexes is 1/4 i

Answer 2

Answer:

Step-by-step explanation:

To multiply complex numbers, we can use the formula:

(a + bi)(c + di) = (ac - bd) + (ad + bc)i,

where "a" and "b" are the real and imaginary parts of the first complex number, and "c" and "d" are the real and imaginary parts of the second complex number.

Let's calculate the multiplication:

First, let's convert the polar form of the second complex number into rectangular form:

1/8 (cos(5π/3) + i sin(5π/3)) = 1/8 (-1/2 + i √3/2) = (-1/16) + (i √3/16).

Now, we can multiply the two complex numbers:

(-√3 + i) * (-1/16 + i √3/16) = (-√3 * (-1/16) - (i * i √3/16)) + (i * (-√3 * (1/16)) + (-√3 * i √3/16))

= (√3/16 + 3i/16) + (-3i√3/16 - √3/16)

= (√3/16 - √3/16) + (3i/16 - 3i√3/16)

= 0 + (3i - 3i√3)/16

= (3i - 3i√3)/16.

Therefore, the result of multiplying the complex numbers -√3 + i and 1/8 (cos 5π/3 + i sin 5π/3) is (3i - 3i√3)/16.

which can also be written as (1/4)i