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4 years ago

Solve for x. (e^x-e^-x)/((e^x+e^-x)=t

2 answers:

6 0

$\dfrac{e^x-e^{-x}}{e^x+e^{-x}}=t\\\\\dfrac{e^x-e^{-x}}{e^x+e^{-x}}=(e^x-e^{-x}):(e^x+e^{-x})=\left(e^x-\dfrac{1}{e^x}\right):\left(e^x+\dfrac{1}{e^x}\right)\\\\=\left(\dfrac{(e^x)^2}{e^x}-\dfrac{1}{e^x}\right):\left(\dfrac{(e^x)^2}{e^x}+\dfrac{1}{e^x}\right)=\dfrac{e^{2x}-1}{e^x}:\dfrac{e^{2x}+1}{e^x}\\\\=\dfrac{e^{2x}-1}{e^x}\cdot\dfrac{e^x}{e^{2x}+1}=\dfrac{e^{2x}-1}{e^{2x}+1}\\\\\text{substitute}\ e^{2x}=s$

$\text{therefore}\\\\\dfrac{e^x-e^{-x}}{e^x+e^{-x}}=t\to\dfrac{s-1}{s+1}=t\ \ \ \ |\cdot(s+1)\\\\s-1=t(s+1)\\\\s-1=ts+t\ \ \ \ |+1\\\\s=ts+t+1\ \ \ |-ts\\\\s-ts=t+1\\\\s(1-t)=t+1\ \ \ \ |:(1-t)\neq0\\\\s=\dfrac{t+1}{1-t}\to e^{2x}=\dfrac{t+1}{1-t}\ \ \ \ |\ln\\\\\ln e^{2x}=\ln\left(\dfrac{t+1}{1-t}\right)\\\\2x\ln e=\ln\left(\dfrac{t+1}{1-t}\right)\\\\2x=\ln\left(\dfrac{t+1}{1-t}\right)\ \ \ \ |:2\\\\\boxed{x=\dfrac{1}{2}\ln\left(\dfrac{t+1}{1-t}\right)}\\\\\text{The domain:}\\\\t\in(-1;\ 1)$

$\text{Used:}\\\\a^{-n}=\dfrac{1}{a^n}\\\\(a^n)^m=a^{nm}\\\\\log_ab^n=n\cdot\log_ab\\\\\log_aa=1$

SIZIF [17.4K]4 years ago

4 0

Answer:

its d on edge ;)

Step-by-step explanation:

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The gandhara region is the home of many portrayals of ____.

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BUDDHA. It is a cold climate region where many BUDDHA carvings, with simplistic garments and appearance can be seen. This also exemplifies Gandhara art work that is usually influenced by roman empire.

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3 years ago

In triangle abc, cos A= -0.6. Find A and Tan A

Tresset [83]

Answer:

Part 1) $A=126.87^o$

Part 2) $tan(A)=-\frac{4}{3}$

Step-by-step explanation:

we have

$cos(A)=-0.6$

The cos(A) is negative, that means that the angle A in the triangle ABC is an obtuse angle and the value of the sin(A) is positive

The angle A lie on the II Quadrant

step 1

Find the measure of angle A

$cos(A)=-0.6$

using a calculator

$A=cos^{-1}(-0.6)=126.87^o$

step 2

Find the sin(A)

we know that

$sin^2(A)+cos^2(A)=1$

substitute the value of cos(A)

$sin^2(A)+(-0.6)^2=1$

$sin^2(A)=1-0.36$

$sin^2(A)=0.64$

$sin(A)=0.8$

step 3

Find tan(A)

we know that

$tan(A)=\frac{sin(A)}{cos(A)}$

substitute the values

$tan(A)=\frac{0.8}{-0.6}$

Simplify

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3 years ago

The Thomas family bought 5 subs and 4 drinks. They spent 49.50 write and equation that models this situation define x and y vari

OlgaM077 [116]

Answer:

5x+4y=49.50, where:

x is the price per sub

y is the price per drink

Step-by-step explanation:

From the information provided, you can say that the result of adding up the price per sub for the quantity plus the price per drink for the quantity is equal to 49.50 that is the total amount spent, which can be expressed as:

5x+4y=49.50, where:

x is the price per sub

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3 years ago

There are 750 toothpicks in a regular sized box. If a jumbo box is made by doubling all the dimensions of the regular sized box,

Serga [27]

To answer this, let the 3 sides of the regular sized rectangular box be x, y, and z.

When we double all the sides, the new sides have lengths 2x, 2y, and 2z, respectively.

Since the regular box (with dimensions x,y, and z) have 750 toothpicks, we can write

$xyz=750$