Write the polar form of z=1+cosa+i*sina.

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

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The complex number z can be represented by the plane polar coordinates (r, a).

z = r(cos theta+ i*sin theta)

We know, from enunciation, the original form of the given complex  number:

z=1+cosa+i*sina

Re(z) = 1 + cos a

Im (z) = sin a

We'll write 1 + cos a = 2 [cos (a/2)]^2

sin a = 2 sin(a/2)*cos (a/2)

We'll re-write z:

z = 2 [cos (a/2)]^2 + i*2 sin(a/2)*cos (a/2)

We'll factorize by 2cos (a/2):

z = 2cos (a/2)[cos (a/2) + sin(a/2)*i]

Now, we'll identify the polar coordinates r and a:

r = 2cos (a/2)

theta = arg (z) = (a/2)

The polar form of z is:

z = 2cos (a/2)[cos (a/2) + sin(a/2)*i]

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neela | High School Teacher | (Level 3) Valedictorian

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z = 1+cosa +isina..

The polar form of z = r( cosx +isinx).

Therefore r cosx = 1+cosa and

 r sinx = sina.

 (rcosx)^2 +(rsinx)^2 = (1+cosa)^2+ (sina)^2

r^2 = 1+2cosa +cos^2a+sin^2a = 1+2cosa+1= 2(1+cosa).

Therefore r = (sqrt2) sqrt(1+cosa) = sqrt2 {sqrt{1+2cos^2(a/2)-1} = 2cos(a/2).

Therefore cosx = (1+cosa)/sqrt2sqrt(1+cosa) = (1/2)(sqrt2)sqrt(1+cosa)

sinx = (1/2)(sqrt2) (sina)/ sqrt(1+cosa).

Tanx = sinx/cosx = sina/(1+cosa) = 2(sina/2)(cosa/2)/{1+1-2sin^2(a/2)} =   2(sina/2 )(cosa/2)/2{1-sin^2(a/2)} = tan(a/2).

Therefore x = a/2.

Therefore cosx = cosa/2 and sinx = sina/2.

Therefore  the polar form  of z = 1+cosa +isina = 2cos(a/2){ cos(a/2)+ isin(a/2)}.

 

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