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Thepotemich [5.8K]
3 years ago
9

Envision geometry 2018 question 1.3.9 answer

Mathematics
1 answer:
kozerog [31]3 years ago
8 0

Answer:

Jajajja

Step-by-step explanation:

Nejsjs

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It would be two times the angle ABC


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A. 20<br> b. 5<br> c. 40<br> d. 45
Rina8888 [55]

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C, 40

Step-by-step explanation:

You can set up the equation as 20+4x=180

180 is the total angle of the line

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Addition

Step-by-step explanation:

1) Add 275 to both sides.

800 + 275 = p

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The Flushing River has a current of 3mph. A motorboat takes as long to go 12 miles downstream as to go 8 miles upstream. What is
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2 years ago
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.
6 0
3 years ago
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