Hyperbolic functions and algebraic methods

[Denton91]

I have done most of my coursework, but I don't even know where to start on the following question:
Solve the equation by algebraic methods: 4.6cosh(x)+3.2sinh(x)=10

i don't know whether it is just brain freeze or what, but I don't know where to start, or even find anywhere on the internet to point me in the right direction.

thanks in advance

 

[HallsofIvy]

Do you know what "hyperbolic functions" are?

They are defined as \(\displaystyle cosh(x)= \frac{e^x+ e^{-x}}{2}\) and \(\displaystyle sinh(x)= \frac{e^x- e^{-x}}{2}\) with the other functions, tanh(x), coth(x), sech(x), csch(x) defined analogously to the corresponding trig functions.

So your equation, \(\displaystyle 4.6cosh(x)+3.2sinh(x)=10\) is the same as \(\displaystyle 2.3(e^x+ e^{-x})+ 1.6(e^x- e^{-x})= 5.8e^x- 2.6e^{-x}= 10\).

Multiply through by \(\displaystyle e^x\) to get \(\displaystyle 5.8 e^{2x}- 2.6= 10e^{x}\). Let \(\displaystyle y= e^x\) and you have a quadratic equation to solve for y.

 

[stapel]

I have done most of my coursework, but I don't even know where to start on the following question:
Solve the equation by algebraic methods: 4.6cosh(x)+3.2sinh(x)=10

Maybe start with the definitions of these hyperbolic trig functions, and go from there? (I haven't done all the computations, so this is only an "off the top of my head" suggestion!)

. . . . .\(\displaystyle \displaystyle{ \left(\frac{4.6}{2}\right)\left(e^x\, +\, e^{-x}\right)\, +\, \left(\frac{3.2}{2}\right)\left(e^x\, -\, e^{-x}\right)\, =\, 10 }\)

. . . . .\(\displaystyle \displaystyle{ 2.3\left(e^x\, +\, e^{-x}\right)\, +\, 1.6\left(e^x\, -\, e^{-x}\right)\, =\, 10 }\)

. . . . .\(\displaystyle \displaystyle{ 2.3e^x\, +\, 1.6e^x\, +\, 2.3e^{-x}\, -\, 1.6e^{-x}\, =\, 10 }\)

...and so forth. Multiply through by \(\displaystyle e^x\) to get a quadratic in \(\displaystyle e^x\), and then solve the quadratic for solutions in the form of exponential equations. Solve these to find the answer to the original exercise.

 

[Denton91]

I have no idea about hyperbolic functions until I searched the internet today for help with this question. My tutor teaches in a way that he thinks we should already know everything, and is hardly interested in helping. Thanks for your help, I will give it a go again, wish me luck

 

[Deleted member 4993]

I have done most of my coursework, but I don't even know where to start on the following question:
Solve the equation by algebraic methods: 4.6cosh(x)+3.2sinh(x)=10

i don't know whether it is just brain freeze or what, but I don't know where to start, or even find anywhere on the internet to point me in the right direction.

thanks in advance

Another way......

I'll show you a different but similar problem.

Solve

A*cos(Θ) + B*sin(Θ) = C

A/√(A² + B²)*cos(Θ) + B/√(A² + B²)*sin(Θ) = C/√(A² + B²)

Assuming |C| ≤ √(A² + B²) and

cos(Φ) = C/√(A² + B²) and

cos(ß) = A/√(A² + B²)

sin(ß) = B/√(A² + B²)

Then

A/√(A² + B²)*cos(Θ) + B/√(A² + B²)*sin(Θ) = C/√(A² + B²) turns into

cos(ß)*cos(Θ) + sin(ß)*sin(Θ) = cos(Φ)

cos(ß - Θ) = cos(Φ) → ß - Θ = 2 * k * π ± Φ

Find similar relationships between cosh(x) and sin(hx).

To start with:

You know → cos²(Θ) + sin²(Θ) = 1

the corresponding hyperbolic equation is: → cosh²(x) - sinh²(x) = 1

You know

cos(A+B) = cos(A)*cos(B) - sin(A)*sin(B)

and

cos(A+B) = cos(A)*cos(B) - sin(A)*sin(B)

Find (calculate) corresponding equations in the hyperbolic domain and continue......

But I like Staple's method much better!!!

 

[Denton91]

Another way......I'll show you a different but similar problem.SolveA*cos(Θ) + B*sin(Θ) = CA/√(A² + B²)*cos(Θ) + B/√(A² + B²)*sin(Θ) = C/√(A² + B²)Assuming |C| ≤ √(A² + B²) and cos(Φ) = C/√(A² + B²) andcos(ß) = A/√(A² + B²)sin(ß) = B/√(A² + B²)ThenA/√(A² + B²)*cos(Θ) + B/√(A² + B²)*sin(Θ) = C/√(A² + B²) turns intocos(ß)*cos(Θ) + sin(ß)*sin(Θ) = cos(Φ)cos(ß - Θ) = cos(Φ) → ß - Θ = 2 * k * π ± ΦFind similar relationships between cosh(x) and sin(hx).To start with:You know → cos²(Θ) + sin²(Θ) = 1the corresponding hyperbolic equation is: → cosh²(x) - sinh²(x) = 1You know cos(A+B) = cos(A)*cos(B) - sin(A)*sin(B) and cos(A+B) = cos(A)*cos(B) - sin(A)*sin(B)Find (calculate) corresponding equations in the hyperbolic domain and continue......But I like Staple's method much better!!!

Thanks everyone for the help, I managed to get it done, seems so easy now...but coming up with the answer, I ended up with 2 (quadratic equation) a positive number and a negative number. Is the negative number a viable answer as an alternative to the positive, or is it just the positive?Also, thank you Subhotosh...for an higher mark, i need an alternative method to answer the question...i will try get my head around your method some time tonight

 

[Deleted member 4993]

Thanks everyone for the help, I managed to get it done, seems so easy now...but coming up with the answer, I ended up with 2 (quadratic equation) a positive number and a negative number. Is the negative number a viable answer as an alternative to the positive, or is it just the positive?Also, thank you Subhotosh...for an higher mark, i need an alternative method to answer the question...i will try get my head around your method some time tonight

What is the range of the function y = e^x ?

 

[Denton91]

What is the range of the function y = e^x ?

When '+' the Q.E I was left with ex=2.4921, when '-' the Q.E I was left with ex=0.0720, which leads me to believe that both therefore will result in correct answers

 

[HallsofIvy]

Yes, both of those are correct. Since those are values for e^x, what is x?

 

[Denton91]

Yes, both of those are correct. Since those are values for e^x, what is x?

I ended up with x=0.9131 and x=-2.6311. I also plotted these onto a graph as a 2nd method. Only need to find a second method to two more questions now :cool: