Determine the trigonometric form of the number z=1

Expert Answers
hala718 eNotes educator| Certified Educator

z= 1

Let z= 1+bi

==> a + bi = 1

But we know that:

z= l zl (cos t + (sin t)*i)

But,  lzl = sqrt(1) = 1

==> 1= 1* cost + sint* i

==> 1 + 0i = cost + sint*i

==> cost = 1

==> sint = 0

==> t= 0

==> z= cos0 + sin0 *i

giorgiana1976 | Student

The given complex number z, is written in the algebraic form:

z = a+b*i, where a is the real part and b is the imaginary part.

z = 1 + 0*i

Re(z) = 1

The complex number written in trigonometric form is:

z = |z|*(cos t+ i*sin t)

where:

|z| = sqrt (a^2 + b^2)

cos t = a / |z| and sint  = b / |z|

We'll calculate |z|:

|z| = sqrt(1^2 + 0^2)

|z| = 1

cos t = a / |z|

cos t = 1/ 1

cos t = 1 => t = 0

sin t = b / |z|

sin t = 0/1

sin t = 0 => t = 0

We'll put z in trigonometric form:

z = 1*(cos 0 + i*sin 0)

z = (cos 0 + i*sin 0)

neela | Student

z =1 To find the trigonometric form .

Solution:

We know that z = a+bi  in the Argand plane, where a and b are real.

So z =  1+ i*0 = cos0+i*sin0, as cos 0 =1 and sin0 = 0.

Also z = cos(2npi)+i sin(2npi), as co2npi = 1 and sin2npi = o for n =0,1,2,3.....

 

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