Question

Express 2+6√3i/5+√3i in polar form

Answer 1

Answer:

2 {cos(π/3) + i sin(π/3)}

Solution:

Let, z = (2 + 6√3i) / (5 + √3i)

= {(2 + 6√3i) (5 - √3i)}/{(5 + √3i) (5 - √3i)}

= (10 - 2√3i + 30√3i - 18i²)/(25 - 3i²)

= (10 + 28√3i + 18) / (25 + 3),

where i = √(- 1)

= (28 + 28√3i)/28

= 1 + i√3

Let, 1 + i√3 = r (cosθ + sinθ)

Then r cosθ = 1 and r sinθ = √3

Now, r² cos²θ + r² sin²θ = 4

Then r² = 4

So, r = 2

Then cosθ = 1/2 and sinθ = √3/2

This gives θ = π/3

Therefore mod z = 2 and arg z = π/3

Hence 1 + i√3 = 2 {cos(π/3) + i sin(π/3)}

Answer 2

Answer:

see the answer is present in target publication guide part 2

Step-by-step explanation: