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alexandr402 [8]
2 years ago
9

20 points and brainliest I’m in quiz in need it asap Number 4

Mathematics
1 answer:
iren [92.7K]2 years ago
7 0

Answer and step-by-step explanation:

The polar form of a complex number a+ib is the number re^{i\theta} where r = \sqrt{a^2+b^2} is called the modulus and \theta = tan^-^1 (\frac ba) is called the argument. You can switch back and forth between the two forms by either remembering the definitions or by graphing the number on Gauss plane. The advantage of using polar form is that when you multiply, divide or raise complex numbers in polar form you just multiply modules and add arguments.

(a) let's first calculate moduli and arguments

r_1 = \sqrt{(-2\sqrt3)^2+2^2}=\sqrt{12+4} = 4\\ \theta_1 = tan^-^1(\frac{2}{-2\sqrt3}) =-\pi/6\\r_2=\sqrt{1^2+1^2}=\sqrt2\\ \theta_2 = tan^-^1(\frac 11)= \pi/4

now we can write the two numbers as

z_1=4e^{-i\frac \pi6}; z_2=e^{i\frac\pi4}

(b) As noted above, the argument of the product is the sum of the arguments of the two numbers:

Arg(z_1\cdot z_2) = Arg(z_1)+Arg(z_2) = -\frac \pi6 + \frac \pi4 = \frac\pi{12}

(c) Similarly, when raising a complex number to any power, you raise the modulus to that power, and then multiply the argument for that value.

(z_1)^1^2=[4e^{-i\frac \pi6}]^1^2=4^1^2\cdot (e^{-i\frac \pi6})^1^2=2^2^4\cdot e^{-i(12)\frac\pi6}\\=2^2^4 e^{-i\cdot2\pi}=2^2^4

Now, in the last step I've used the fact that e^{i(2k\pi+x)} = e^i^x ; k\in \mathbb Z, or in other words, the complex exponential is periodic with 2\pi as a period, same as sine and cosine. You can further compute that power of two with the help of a calculator, it is around 16 million, or leave it as is.

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I need help on 17 and 23 pls
ludmilkaskok [199]

                           

                          Question # 17 Solution

Answer:

x_{1}= \frac{mx_{2}-y_{2}+y_{1}}{m}

Step-by-step Explanation:

The given expression

m = \frac{y_{2}-y_{1}}{x_{2}-x_{1}}

And we have to solve for x₁

So,

Lets solve for x₁.

m = \frac{y_{2}-y_{1}}{x_{2}-x_{1}}

Multiply both sides by x₂ - x₁

m(x_{2}-x_{1})= (x_{2}-x_{1})\frac{y_{2}-y_{1}}{x_{2}-x_{1}}

m(x_{2}-x_{1})= (y_{2}-y_{1})

mx_{2}-mx_{1}= (y_{2}-y_{1})

-mx_{1}= y_{2}-y_{1}-mx_{2}

Divide both sides by -m

\frac{-mx_{1}}{-m}= \frac{y_{2}-y_{1}-mx_{2}}{-m}

x_{1}= \frac{mx_{2}-y_{2}+y_{1}}{m}

Therefore, x_{1}= \frac{mx_{2}-y_{2}+y_{1}}{m}

                                     Question # 23 Solution

Answer:

n= \frac{bx}{-b + x}

Step-by-step Explanation:

The given expression

\frac{nx}{b}-x=x

And we have to solve for n

So,

Let's solve for n.

\frac{nx}{b}-x=x

Multiply both sides by b.

-bn + nx = bx

Factor out n.

n(-b + x)= bx

Divide both sides by -b + x.

\frac{n(-b + x)}{-b + x}= \frac{bx}{-b + x}

n= \frac{bx}{-b + x}

Therefore, n= \frac{bx}{-b + x}

<em>Keywords: solution, equation</em>

<em>Learn more about the solution of equations from brainly.com/question/12864981</em>

<em>#learnwithBrainly</em>

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set both p equal:
x+38=7x-10
38=6x-10
48=6x
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then insert x=8 in all the side/perimeter equations:
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sides:
x=8
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x+3=8+3=11
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