Given the complex number Z = (1 + cos @ + (1 - Sin @) I, then the maximum value of Z is? Which requires a clear process

Given the complex number Z = (1 + cos @ + (1 - Sin @) I, then the maximum value of Z is? Which requires a clear process

|z|=(1+cos a)²+(1-sin a)²=1+2cosa+cos²a+1-2sina+sin²a=3+2√2cos(a+π/4)
It can be seen that when a + π / 4 = 2K π, | Z | is 3 + 2 √ 2 at most

Find the value range of modulus of complex z = (1-cos θ) + (2 + sin θ) I

Z-module = √ [(1-cos θ) ^ 2 + (2 + sin θ) ^ 2] = √ (1-2cos θ + cos ^ 2 θ + 4 + 4sin θ + sin ^ 2 θ) = √ [6 + 2 (2Sin θ - cos θ)] = √ [6 + 2 * √ 5sin (θ - φ)]. Auxiliary angle formula, Tan φ = 1 / 2  maximum value of module = √ (6 + 2 √ 5) = √ 5 + 1 minimum value = √ (6-2 √ 5) = √ 5-1  value range [√ 5

Given the complex z = (3 + cos θ) + (- 1-sin θ) I, then the locus of the point on the complex plane corresponding to the complex Z is

X = 3 + cos θ y = - 1-sin θ, then (x-3) ^ 2 + (y + 1) ^ 2 = 1, that is, the circle with the center of (3, - 1) and radius of 1