express 1- sin α + i cos α in the form of r (cos θ + i sin θ).

Question

Philoid · Accepted Answer

Given Complex number is z = 1 – sinα +I cosα

We know that,

sin²θ + cos²θ = 1,

sin 2θ = 2 sin θ cos θ,

cos 2θ = cos²θ - sin²θ

⇒

We know that the polar form of a complex number Z = x + iy is given by Z=|Z|(cos θ + I sin θ)

Where,

|Z|=modulus of complex number=

θ =arg(z)=argument of complex number=

⇒

We know that sine and cosine functions are periodic with period 2

Here We have 3 intervals as follows:

(i)

(ii)

(iii)

Case(i):

In the interval , and also

so,

⇒

⇒ .(∵ θ lies in 1^st quadrant)

∴ The polar form is .

Case(ii):

In the interval , and also

so,

⇒

⇒ . (∵ θ lies in 4^th quadrant)

⇒

∴ The polar form is .

Case(iii):

In the interval , and also

so,

⇒

⇒ .(since θ presents in first quadrant and tan’s period is )

⇒ .

∴ The polar form is .