Question

23. Find modulus & argument of z=1+i​

Answer 1

Answer-

Argument = π / 4

Modulus = $\sqrt{2$

Explanation :

z= rCosФ + i SinФ

z= 1+i

by comparing both , we can say ,

r CosФ = 1  ,  r SinФ = 1

squaring and adding the above two results ,

$r^{2} Cos^2$Ф + $r^2 Sin^2$Ф = $1^2 + (-1)^2$

$r^2 ( Cos^2$Ф+ $Sin^2$Ф) = 1+1

$r^2$(1) = 2              {bcz , $Sin^2$Ф+$Cos^2$Ф =1}

r=$\sqrt{2}$

So, modulus = $\sqrt{2}$

Now, divide the two results given above ,

r SinФ / r Cos Ф = 1 / 1

Sin Ф / Cos Ф = 1

Tan Ф = 1 = $\pi$ / 4

Here , both SinФ and Cos Ф are positive , so Ф lies in first quadrant .

Therefore, Ф=π/4

So, argument = π / 4