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4 years ago

Help NEEDED!!!! plzzzzzz.....

Mathematics

1 answer:

julsineya [31]4 years ago

7 0

Sin30= opposite/hypotenuse
sin30= AC/10
AC= 4.5cm
Tan25= opposite/adjacent
Tan25= 4.5/ CD
CD= 10.7

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How to prove tan z is analytic using cauchy-riemann conditions

Basile [38]

A function $f(z)=f(x+iy)=u(x,y)+i v(x,y)$ is analytic if the C-R conditions are satisfied:

$\begin{cases}u_x=v_y\\u_y=-v_x\end{cases}$

We have

$f(z)=\tan z=\tan(x+iy)$

Recall the angle sum formula for tangent:

$\tan(x+iy)=\dfrac{\tan x+\tan iy}{1-\tan x\tan iy}$

Now recall that

$\tan iy=\dfrac{\sin iy}{\cos iy}=\dfrac{-\frac{e^y-e^{-y}}{2i}}{\frac{e^y+e^{-y}}2}=\dfrac{i\sinh y}{\cosh y}=i\tanh y$

So we have

$\dfrac{\tan x+\tan iy}{1-\tan x\tan iy}=\dfrac{\tan x+i\tanh y}{1-i\tan x\tanh y}$
$=\dfrac{(\tan x+i\tanh y)(1+i\tan x\tanh y)}{(1-i\tan x\tanh y)(1+i\tan x\tanh y)}$
$=\dfrac{\tan x+i\tanh y+i\tan^2x-\tan x\tanh^2y}{1+\tan^2x\tanh^2y}$
$=\dfrac{\tan x(1-\tanh^2y)}{1+\tan^2x\tanh^2y}+i\dfrac{\tanh y(1+\tan^2x)}{1+\tan^2x\tanh^2y}$
$=\underbrace{\dfrac{\tan x\sech^2y}{1+\tan^2x\tanh^2y}}_{u(x,y)}+i\underbrace{\dfrac{\tanh y\sec^2x}{1+\tan^2x\tanh^2y}}_{v(x,y)}$

We could stop here, but taking derivatives may be messy and would be easier to do if we can write this in terms of sines and cosines.

$u(x,y)=\dfrac{\tan x\sech^2y}{1+\tan^2x\tanh^2y}=\dfrac{\frac{\sin x}{\cos x\cosh^2x}}{1+\frac{\sin^2x\sinh^2y}{\cos^2x\cosh^2y}}=\dfrac{\sin x\cos x}{\cos^2x\cosh^2y+\sin^2x\sinh^2y}$
$u(x,y)=\dfrac{\sin2x}{2\cos^2x\cosh^2y+2(1-\cos^2x)(\cosh^2y-1)}=\dfrac{\sin2x}{2\cosh^2y-1+2\cos^2x-1}$
$u(x,y)=\dfrac{\sin2x}{\cos2x+\cosh2y}$

With similar usage of identities, we can find that

$v(x,y)=\dfrac{\sinh2y}{\cos2x+\cosh2y}$

Now we check the C-R conditions.

$u_x=\dfrac{2\cos2x(\cos2x+\cosh2y)-(-2\sin2x)\sin2x}{(\cos2x+\cosh2y)^2}=\dfrac{2+2\cos2x\cosh2y}{(\cos2x+\cosh2y)^2}$
$v_y=\dfrac{2\cosh2y(\cos2x+\cosh2y)-(2\sinh2y)\sinh2y}{(\cos2x+\cosh2y)^2}=\dfrac{2+2\cosh2y\cos2x}{(\cos2x+\cosh2y)^2}$
$\implies u_x=v_y$

Similarly, you can check that $u_y=-v_x$ , hence the C-R conditions are satisfied, and so $\tan z$ is analytic.

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If i ate 20 candy bars and 40 more what do i have

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Answer:

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Step-by-step explanation:

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3 years ago

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An online furniture store sells chairs for $200 each and tables for $600 each. Every day, the store can ship a maximum of 32 pie

wolverine [178]

Answer:

If the minimum 13 chair were sold , then the minimum number of Table sold is 14

Step-by-step explanation:

Given as :

The cost of each chair = $ 200

The cost of each table = $ 600

The total number of furniture sold per day = 32

The minimum amount of selling per day = $ 12000

Let the total number of chair = C

The total number of table = T

So , according to question

C + T = 32 .......1

200 C + 600 T = 12000 .......2

Solving eq 1 and 2

200 C + 600 T = 12000

200 × ( C + T ) = 32 × 200

I.e 200 C + 200 T = 6400

or, ( 200 C + 600 T ) - ( 200 C + 200 T ) = 12,000 - 6400

Or, 400 T = 5600

∴ T = $\frac{5600}{400}$

I.e T = 14

Put The value of T in eq 2

So, 200 C + 600 × 14 = 12000

or , 200 C + 8400 = 12,000

Or, 200 C = 12000 - 8400

Or, 200 C = 3600

∴ C = $\frac{3600}{200}$

I.e C = 18

The number of chair sold is 18

If the number of chair sold is 13 ,

Then the min number of table sold = 200 × 18 + 600 T = 12000

i.e 600 T = 12000 - 3600

or, 600 T = 8400

∴ T = $\frac{8400}{600}$

I.e T = 14

Hence if the minimum 13 chair were sold , then the minimum number of Table sold is 14 . Answer

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