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4 years ago

Prove cosh 3x = 4 cosh^3 x - 3 cosh x.

1 answer:

3 0

Prove we are to prove 4(coshx)^3 - 3(coshx) we are asked to prove 4(coshx)^3 - 3(coshx) to be equal to cosh 3x
= 4(e^x+e^(-x))^3/8 - 3(e^x+e^(-x))/2 = e^3x /2 +3e^x /2 + 3e^(-x) /2 + e^(-3x) /2 - 3(e^x+e^(-x))/2 = e^(3x) /2 + e^(-3x) /2 = cosh(3x) = LHS Since y = cosh x satisfies the equation if we replace the "2" with cosh3x, we require cosh 3x = 2 for the solution to work.
i.e. e^(3x)/2 + e^(-3x)/2 = 2
Setting e^(3x) = u, we have u^2 + 1 - 4u = 0
u = (4 + sqrt(12)) / 2 = 2 + sqrt(3), so x = ln((2+sqrt(3))/2) /3, Or u = (4 - sqrt(12)) / 2 = 2 - sqrt(3), so x = ln((2-sqrt(3))/2) /3,
Therefore, y = cosh x = e^(ln((2+sqrt(3))/2) /3) /2 + e^(-ln((2+sqrt(3))/2) /3) /2 = (2+sqrt(3))^(1/3) / 2 + (-2-sqrt(3))^(1/3) to be equ
= 4(e^x+e^(-x))^3/8 - 3(e^x+e^(-x))/2
= e^3x /2 +3e^x /2 + 3e^(-x) /2 + e^(-3x) /2 - 3(e^x+e^(-x))/2
= e^(3x) /2 + e^(-3x) /2
= cosh(3x)
= LHS

Therefore, because y = cosh x satisfies the equation IF we replace the "2" with cosh3x, we require cosh 3x = 2 for the solution to work. 

i.e. e^(3x)/2 + e^(-3x)/2 = 2

Setting e^(3x) = u, we have u^2 + 1 - 4u = 0

u = (4 + sqrt(12)) / 2 = 2 + sqrt(3), so x = ln((2+sqrt(3))/2) /3,
Or u = (4 - sqrt(12)) / 2 = 2 - sqrt(3), so x = ln((2-sqrt(3))/2) /3,

Therefore, y = cosh x = e^(ln((2+sqrt(3))/2) /3) /2 + e^(-ln((2+sqrt(3))/2) /3) /2
= (2+sqrt(3))^(1/3) / 2 + (-2-sqrt(3))^(1/3)

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What is the value of 24/4 + (-2×5)

zhenek [66]

24/2 + (-2x5) = 12 + (-10) = 2

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3 years ago

What are the x- and y- coordinates of point E, which partitions the directed line segment from J to K into a ratio of 1:4? (–13,

Lana71 [14]

I added a screenshot of the complete question.

Answer:

(-7, -1)

Explanation:

The formulas given to calculated the x and y coordinates are as follows:

$x = (\frac{m}{m+n})(x_{2}-x_{1}) + x_{1}\\\\y = (\frac{m}{m+n})(y_{2}-y_{1}) + y_{1}$

Let's define the variables used:

(x₁ , y₁) are the coordinates of the first point while (x₂ , y₂) are the coordinates of the second point.

We are given that the segment is directed from J to K, therefore:

First point is J ..........> (x₁ , y₁) is (-15, -5)

Second point is K ....> (x₂ , y₂) is (25, 15)

m and n defined the portion of the partitioned segment (JE : EK). It is given that this ratio is 1:4. Therefore:

m = 1 and n = 4

Finally, let's substitute with these variables in the equations as follows:

$x = (\frac{1}{1+4})(25-(-15)) + (-15) = -7\\\\y = (\frac{1}{1+4})(15-(-5)) + (-5) = -1$

Based on the above, the coordinates of point E are (-7, -1)

Hope this helps :)

8 0

4 years ago

Read 2 more answers

An airplane is traveling at a speed of 60 meters per second. Identify the correct conversion factor setup required to compute th

Nezavi [6.7K]

Answer:

To convert m/sec to km/h the factor is 3.6

The speed of the airplane is $216\frac{km}{h}$

Step-by-step explanation:

we know that

$1\ km=1,000\ m\\1\ h=3,600\ sec$

$60\frac{m}{sec}=60\frac{(1/1,000)}{(1/3,600)}=60\frac{3,600}{1,000}=216\frac{km}{h}$

To convert m/sec to km/h the factor is 3.6

That means ----> Multiply the speed in m/sec by 3.6 to obtain the speed in km/h

7 0

3 years ago

What is the surface area of this square pyramid?

Karo-lina-s [1.5K]

Answer:

A;3600 yd2

Step-by-step explanation:

The surface area is found by adding the area of the sides

The area of the bottom is

A= s^2 where s is the side length since it is a square

A = s^2 = 30^2= 900

We can find the area of one of the sides and multiply by 4 since all the sides are the same

The sides are triangles so

A = 1/2 bh where b is the base b=30 and h is the height = 45

A = 1/2 (30)* 45

A = 675

We have 4 of these triangles

4*675 =2700

Add the 4 triangles and the bottom

2700+900

3600

6 0

4 years ago

A recently admitted class of graduate students at a large state university has a mean GRE verbal score of 650 with a standard de

hjlf

Answer:

The answer is below

Step-by-step explanation:

The z score is a score used to determine the number of standard deviations by which the raw score is above or below the mean, it is given by the equation:

$z=\frac{x-\mu}{\sigma} \\\\\mu=mean, \sigma=standard\ deviation,x=raw\ score\\\\for\ a\ sample\ n:\\\\z=\frac{x-\mu}{\sigma/\sqrt{n} }$

Given that μ = 650, σ = 50. To find the probability that 5 students who have a mean of 490, we use:

$z=\frac{x-\mu}{\sigma/\sqrt{n} } =\frac{490-650}{50/\sqrt{5} } =-7.16$

From the normal distribution table, P(x < 490) = P(Z < -7.16) = 0.0001 = 0.01%

Since only a small percentage of people score about 490, hence the local newspaper editor should write a scathing editorial about favoritism

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3 years ago