Express the following complex numbers in the form 1 + i tan α

Question

Philoid · Accepted Answer

Given Complex number is Z=1+itanα

We know that the polar form of a complex number Z=x+iy is given by Z=|Z|(cosθ+isinθ)

Where,

|Z|=modulus of complex number=

θ =arg(z)=argument of complex number=

We know that tanα is a periodic function with period .

We have lying in the interval

Case1:

⇒

Since sec is positive in the interval

⇒

Since tan is positive in the interval

⇒ θ=

∴ The polar form is z=sec(cos+isin).

Case2:

⇒

Since sec is negative in the interval .

⇒

Since tan is negative in the interval .

⇒ .(∵ θ lies in 4^th quadrant)

⇒ z=-sec(cos()+isin())

⇒ z=-sec(-cos-isin)

⇒ z=sec(cos+isin)

∴ The polar form is z=sec(cos+isin)