Answer:
RECTANGULAR FORM: z -w = 1- 1i
POLAR FORM: z-w = √2(cos135°+i sin135°)
Step-by-step explanation:
Find the image attached
Given the following complex numbers
z = √2 (cos 45º + i sin 45° )
w = 2(cos 90° + i sin 90).
The complex number in rectangular form is expressed as z = x+iy
(x, y) are the rectangular coordinates
Given
z = √2 (cos 45º + i sin 45° )
z = √2 (1/√2+ ( 1/√2)i )
z = √2(1/√2) + √2(1/√2)i
z = 1 + i .... 1
Also;
w = 2(cos 90° + i sin 90)
w = 2(0 + 1i)
w = 2(0) + 2i
w = 0+2i ....2
Take their difference:
z - w = 1 + i - (0+2i)
z-w = 1+i-0-2i
z-w = 1-0+i-2i
z-w = 1-i
Hence the values that goes into the box is 1.
z-w = 1 - 1i
In Polar form;
Get the modulus of the resulting complex number:
|z-w| = √1²+1²
|z-w| = √2
Get the argument:
[tex]\theta = tan^{-1}\frac{y}{x} \\\theta = tan^{-1}(\frac{-1}{1}) \\\theta = tan^{-1}(-1) \\\theta = -45^0[/tex]
Since tan is negative in the second quadrant, the angle will be 180-45 = 135°
The polar form of the complex umber is expressed as;
z-w = |z-w|(cosθ-isinθ)
z-w = √2(cos135°+i sin135°)
