Consider the numbers z= sqr2 (cos 45º + i sin 45° )and w = 2(cos 90° + i sin 90). Fill in the blanks to complete the statements.

In rectangular form, z - w = 1 - i

In polar form z-w=

= sqr2 (cos + i sin )

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:

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°)

(0,-3) see attached graph

step-by-step explanation:

a. to solve the system, graph the two equations.

x = y+3 becomes y=x-3 has m=1/1 and b=-3. start at (0,-3) on the y-axis and plot a point. then move up 1 unit and 1 unit to the right. plot a point and connect.

y=-4x+3 has m =-4/1 and b=-3. start at (0,-3) on the y-axis and plot a point. then move down four units and to the right 1 unit. plot a point and connect.

the solution is the point (x,y) where the two lines intersect. by the graph below it is (0,-3).

b. to solve by substitution, substitute y = -4x-3 into x= y+3

x= (-4x-3) +3

x=-4x

5x=0

x=0

now substitute x = 0 back in to find y.

y= -4(0)-3

y=0-3

y=-3

solution is (0,-3)

c. use elimination by adding the equations together.

x-y=3

4x+y=-3

5x=0

x=0

then substitute into one equation to find y.

y=-4(0)-3

y=-3