(1) Given two complex numbers z,-r1(cosa + 1 sin a) and z2-r2(cos θ2 + sin θ(2), prove the following formula for the division of complex numbers without using the Quotient Theorem stated in the text equal to the square of the modulus.

The formula for the division of complex numbers is:

z1/z2 = [(r1 cos a - r2 cos θ2) + i(r1 sin a - r2 sin θ2)] / [r2^2 + r1r2(cos(a - θ2) + i sin(a - θ2))]

To divide two complex numbers, we need to multiply the numerator and denominator by the complex conjugate of the denominator. That is, we multiply z1 by r2(cos θ2 - i sin θ2) and z2 by r2(cos θ2 - i sin θ2). This gives us:

z1/z2 = [(r1/r2)cos a - cos θ2 + i((r1/r2)sin a - sin θ2)] / [(r2 cos θ2 - i sin θ2)(cos a + i sin a)]

Next, we simplify the denominator using the identity cos^2θ + sin^2θ = 1:

z1/z2 = [(r1/r2)cos a - cos θ2 + i((r1/r2)sin a - sin θ2)] / [r2(cos a cos θ2 + sin a sin θ2) + i(r2 sin a cos θ2 - r2 cos a sin θ2)]

Now, we multiply the numerator and denominator by the conjugate of the denominator:

z1/z2 = [(r1/r2)cos a - cos θ2 + i((r1/r2)sin a - sin θ2)] / [r2(cos a cos θ2 + sin a sin θ2) + i(r2 sin a cos θ2 - r2 cos a sin θ2)] * [r2(cos a cos θ2 + sin a sin θ2) - i(r2 sin a cos θ2 - r2 cos a sin θ2)] / [r2(cos a cos θ2 + sin a sin θ2) - i(r2 sin a cos θ2 - r2 cos a sin θ2)]After simplifying, we get:

z1/z2 = [(r1 cos a - r2 cos θ2) + i(r1 sin a - r2 sin θ2)] / [r2^2 + r1r2(cos(a - θ2) + i sin(a - θ2))].

For such more questions on Complex numbers:

https://brainly.com/question/29747482

#SPJ11

We have proved the formula for the division of complex numbers without using the Quotient Theorem, which states that:

z1/z2 = |z1|/|z2| * (cos (θ1 - θ2) + i sin (θ1 - θ2))

To prove the formula for the division of complex numbers without using the Quotient Theorem, we can use the polar form of complex numbers and some trigonometric identities.

Let z1 = r1(cos θ1 + i sin θ1) and z2 = r2(cos θ2 + i sin θ2) be two complex numbers in polar form.

Then, we can write the division of z1 by z2 as:

z1/z2 = r1(cos θ1 + i sin θ1) / r2(cos θ2 + i sin θ2)

Multiplying the numerator and denominator by the conjugate of z2, we get:

z1/z2 = r1(cos θ1 + i sin θ1) / r2(cos θ2 + i sin θ2) * r2(cos θ2 - i sin θ2) / r2(cos θ2 - i sin θ2)

Simplifying the numerator, we get:

z1/z2 = r1r2(cos θ1 cos θ2 + sin θ1 sin θ2 + i(sin θ1 cos θ2 - cos θ1 sin θ2))

Using the identities cos (θ1 - θ2) = cos θ1 cos θ2 + sin θ1 sin θ2 and sin (θ1 - θ2) = sin θ1 cos θ2 - cos θ1 sin θ2, we can write:

z1/z2 = r1r2(cos (θ1 - θ2) + i sin (θ1 - θ2))

Now, we can write z1/z2 in polar form as:

z1/z2 = |z1/z2| (cos φ + i sin φ)

where |z1/z2| = r1/r2 and φ = θ1 - θ2.

We can also see that the relation R does not have the comparability property, since for some complex numbers z1 and z2, it is not true that either z1 R z2 or z2 R z1. For example, if z1 = 1 + i and z2 = -1 - i, then z1 R z2 since |z1| < |z2|, but z2 does not R z1 since |z2| < |z1|. Therefore, R is not a total order on the set of complex numbers.

Know more about Quotient Theorem here:

https://brainly.com/question/29046143

#SPJ11

At Jefferson Middle School, eighty-two students were asked which sports they plan to participate in for the coming year. Twenty students plan to participate in track and cross country; six students in cross country and basketball; and eight students in track and basketball. Twelve students plan to participate in all three sports. A total of thirty students plan to participate in basketball, and a total of forty students plan to participate in cross country. Ten students don't plan to participate in any of the three sports.

How many students plan to just participate in track?

3. I NEED HELP ASAP!! PLEASE PUT AND ANSWER AND STEP BY STEP EQUATION!! IF YOU DON'T KNOW THE ANSWER DON'T PUT ANYTHING!!

Write a rule for the linear function shown in the graph.


A. y = 5/3x + 1

B. y = 3x - 1

C. y = 2/5x + 1

D. y = -5/2x - 1

Which phrase describes the algebraic expression 5+w
A: the product of 5 and w
B: the quotſent of 5 and w
C: w more than 5
D: 5 less than w

maths homework please help me

2x3(5-4)/6.2
Algebra

40 POINTS!
3. Suppose a point at (2,3) is translated to (7,-1). What rule describes this translation?
O A. (x, y) + (x + 5, y - 4)
O B.(z,y) → (2 – 5,y + 4)
O c.(x, y) = (x + 9,y - 2)
O D. (x, y) → (2 – 9,y + 2)

If the sum of the interior angles of a polygon is 29160, how many sides does it have?

Please help, will mark brainliest, do not do for points or I will report, explanation please or no brainliest
Multiply.
2y - 1
2y + 1
4y^2 + 1

Which set of ordered pairs (x, y) could represent a linear function?
A = {(3,5), (5,1), (7, -2), (9,-6)}
B = {(2,-5), (5,1), (6,3), (7,6)}
C = {(-4,3), (-2,0), (0, -4), (2, –7)}
D = {(-3,7), (-1,5), (1,3), (3, 1)}

1. Three different scout troops have 12, 24, and 18 members. These troops must march together in a parade in
the same number of columns. What is the greatest possible number of scouts in each column?
9 scouts
b. 6 scouts
c. 8 scouts
d. 4 scouts

como simplificar 9/3​

I don't understand this? help me! Linear functions 9th-grade math

Find x and y. please help​

‼️ PLEASEEE HELPP ‼️

Which expression is equivalent 10x+10
A.2x+5x+4x+5+5
B.2x+8x+1+3+6
C.5x+5x+2+4+4+1
D.2x+9x+5+5
(7th grade) Combing like terms.

What is the value of y when the value of x is 1

k + 11 = -3k- 25 + 2k

Select all ordered pairs that satisfy the function y =
= 8x + 3
A.
(0,3)
B.
(-1, 43)
C.
(0, -45)
D.
(5, 43)

∆ has the coordinates (3, 1), (5, 2), (4, 3). ∆´´´ has coordinates (-1, 2), (1, 3) and (0, 4). Which choice describes the translation of ∆ to ∆´´? A +4 units in the direction of x and -1 units in the direction of y B -4 units in the direction of x and +1 units in the direction of y C +4 units in the direction of x and +1 units in the direction of y D -4 units in the direction of x and -1 units in the direction of y

fractionsss
15/7 = 2 1/7 right?

What expression is equivalent to 1/3 x -1/5

el area de un rectangulo es 84 cm cuadrados si la base mide 9 cm cuanto mide la altura y cual es su perimetro?​

HELP QUICK. By combining three equations, it can be shown that a2+b2=c2. One of the equations is c=e+f. Which of the following could be the other two equations?

What’s 20 times 69?????

1. The Clock
[tex] \sqrt{81} [/tex]
​

Leah puts together gift bags of toys to sell at the flee market. She places exactly 6 toys in each bag. She only sells full bags of toys Leah has 280 toy cars. She can fill 46 toy cars Part B Leah has enough toy trucks to fill 25 gift bags. How many toy cars and toy trucks were put in gift bags to sell at the flee market

Here is a polygon. Use the scale factor of 1/2 determine the new side length for A *

Solve the equation y – 5x = –7 for y.
A. y = 5x – 7
B. y = –5x – 7
C. y = 5x + 7
D. y = –5x + 7

Is 5 a solution to 3x=35

NEED HELP AND RIGHT ANSWER DONT RUSH! REWARD BRAINLIEST