Find the trigonometric form of a complex numberThe algebraic form of the complex number is z=1+i

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to convert the algebraic form `z = 1 + i` into trigonometric form, such that:

`z = sqrt(1^2 + 1^2)(cos (tan^(-1)(1/1)) + i*sin (tan^(-1)(1/1)))`

Replacing `45^o` for `(tan^(-1)(1/1))` yields:

`z = sqrt 2*(cos 45^o + i*sin 45^o)`

Hence, performing the conversion from the algebraic form `z = 1 + i` into trigonometric form yields `z = sqrt 2*(cos 45^o + i*sin 45^o).`

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

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The algebraic form of any complex number is z = x + y*i.

The trigonometric form of a complex number is:

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

|z| = sqrt(x^2 + y^2)

cos a = x/|z|

sin a = y/|z|

We'll identify x and y for te given complex number:

x = Re(z) = 1

y = Im(z)  = 1 (coefficient of i)

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

|z|  =sqrt2

cos a = 1/sqrt2

cos a = sqrt2/2

sin a = 1/sqrt2

sin a = sqrt2/2

sin a = cos a = sqrt2/2

a = pi/4 = 45 degrees

The trigonometric form of the complex number z is:

z = sqrt2*(cos pi/4 + i*sin pi/4)

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