Question

$- 1 - i \sqrt{ 3}$
Modulus and amplitude
class +1
ex.5(c)​

Answer 1

Given :-

- 1 - i√3

To Find :-

Modulus and amplitude of the given .

Used Concepts :-

• | a + ib | = √a² + b²
• Amplitude is also known as arguement .
• Cos²y + Sin²y = 1

Solution :-

- 1 - i√3

⇒ | - 1 - i√3 | = √ ( - 1 )² + ( -√3 )²

⇒ √ 1 + 3

⇒ √4

⇒ 2

Hence , the modulus of the given is 2 .

Now , Let us assume that ,

r cosx = -1           ----- ( i )

r sinx = -√3        ----- ( ii )

Squaring and adding ( i ) and  ( ii ) ,

( r cosx )² + ( r sinx )² = ( - 1 )² + ( - √3 )²

⇒ r² cos²x + r² sin²x = 1 + 3

⇒ r² ( cos²x + sin²x ) = 4

⇒ r² × 1 = 4

⇒ r² = 4

⇒ r = √4 = 2              { ∵ r > 0 }

Now , By ( i ) ,

⇒ r cosx = - 1

⇒ 2 cosx = -1

⇒ cosx = $\frac{-1}{2}$

⇒ x = $cos ^{-1}$ ( -1/2 )

=> x = -2π/3

Hence , The required amplitude and modulus of the given is -2π/3 and 2 .