If tanh(x)=24/25, find the values of the other hyperbolic function at x
I was able to find coth(x)=25/24, but what is sin, cos, csc, and sec?
I would greatly appreciate your help!!
Actually- cosh(x) = (1/2)(e^x + e^-x)
= (1/2)(7 + 1/7) = 25/7, not 24/7
use the definition of tanh(x)
tanh(x) = (e^2x -1)/(e^2x + 1)
(e^2x -1)/(e^2x + 1) = 24/25
25e^2x - 25 = 24e^2x + 24
e^2x = 49
take √
e^x = ±7
then
sinh(x) = (1/2)(e^x - e^-x)
= (1/2)(7 - 1/7)
= (1/2)(48/7) = 24/7
cosh(x) = (1/2)(e^x + e^-x)
= (1/2)(7 + 1/7) = 24/7
You will find the definitions of the hyperbolic trig functions here
http://en.wikipedia.org/wiki/Hyperbolic_function#Standard_algebraic_expressions
Oh, I'd be delighted to help you out! But before we dive into those hyperbolic shenanigans, I must clarify that the hyperbolic functions do not have direct counterparts to the sine (sin), cosine (cos), cosecant (csc), and secant (sec) functions. However, fear not, for we can always express them using the other hyperbolic functions!
Let's explore the relationships, shall we?
1. The hyperbolic sine (sinh) can be expressed as: sinh(x) = (e^x - e^(-x))/2.
So, the hyperbolic sine (sinh) can be found using the given value of tanh(x):
sinh(x) = sqrt(1 + tanh^2(x)) = sqrt(1 + (24/25)^2) = sqrt(625/625 + 576/625) = sqrt(1201/625)
2. The hyperbolic cosine (cosh) can be expressed as: cosh(x) = (e^x + e^(-x))/2.
And, since tanh(x) = sinh(x) / cosh(x), we can find cosh(x):
cosh(x) = 1 / sqrt(1 - tanh^2(x)) = 1 / sqrt(1 - (24/25)^2) = 1 / sqrt(1 - 576/625) = 1 / sqrt(49/625)
= 1 / (7/25) = 25/7.
3. The hyperbolic cosecant (csch) can be expressed as: csch(x) = 1 / sinh(x).
So, csch(x) = 1 / sqrt(1201/625) = sqrt(625/1201).
4. The hyperbolic secant (sech) can be expressed as: sech(x) = 1 / cosh(x).
Thus, sech(x) = 1 / (25/7) = 7/25.
Remember, these values are specific to the given value of tanh(x) = 24/25. Have fun with your hyperbolic adventures!
To find the values of the other hyperbolic functions at x, we first need to recall the definitions of the hyperbolic functions in terms of exponential functions:
sinh(x) = (e^x - e^(-x))/2
cosh(x) = (e^x + e^(-x))/2
tanh(x) = sinh(x)/cosh(x)
cosech(x) = 1/sinh(x)
sech(x) = 1/cosh(x)
coth(x) = 1/tanh(x)
Given that tanh(x) = 24/25, we can use this information to find the values of the other hyperbolic functions:
1. To find sinh(x), we can rearrange the tanh(x) equation:
sinh(x) = tanh(x) * cosh(x)
sinh(x) = (24/25) * cosh(x)
2. To find cosh(x), we can rearrange the tanh(x) equation:
cosh(x) = 1 / tanh(x)
cosh(x) = 1 / (24/25)
cosh(x) = 25/24
3. To find cosech(x), we can use the definition:
cosech(x) = 1 / sinh(x)
cosech(x) = 1 / [(24/25) * cosh(x)]
cosech(x) = 25 / (24 * cosh(x))
4. To find sech(x), we can use the definition:
sech(x) = 1 / cosh(x)
sech(x) = 1 / (25/24)
sech(x) = 24/25
Now, we have the values of the hyperbolic functions at x:
sinh(x) = (24/25) * (25/24) = 1
cosh(x) = 25/24
cosech(x) = 25 / (24 * (25/24)) = 1
sech(x) = 24/25
Please let me know if anything is unclear or if you have any further questions!
To find the values of the other hyperbolic functions at x when given tanh(x) = 24/25, we can use the definitions and identities of hyperbolic functions.
First, let's recall the definitions of the hyperbolic functions:
1. cosh(x) = (e^x + e^(-x)) / 2
2. sinh(x) = (e^x - e^(-x)) / 2
3. tanh(x) = sinh(x) / cosh(x)
4. sech(x) = 1 / cosh(x)
5. csc(x) = 1 / sinh(x)
6. coth(x) = cosh(x) / sinh(x)
Given that tanh(x) = 24/25, we can rearrange the equation from definition 3 to solve for sinh(x):
tanh(x) = sinh(x) / cosh(x)
24/25 = sinh(x) / cosh(x)
Now, let's solve for sinh(x):
sinh(x) = (24/25) * cosh(x)
Using the definition of cosh(x) and substituting the value of sinh(x), we can find sinh(x) in terms of exponential functions:
sinh(x) = (24/25) * ((e^x + e^(-x)) / 2)
Now, we can find the values of the other hyperbolic functions:
1. cosh(x) = (e^x + e^(-x)) / 2
2. sinh(x) = (24/25) * ((e^x + e^(-x)) / 2)
3. sech(x) = 1 / cosh(x)
4. csc(x) = 1 / sinh(x)
5. coth(x) = cosh(x) / sinh(x)
To simplify further, we can calculate the values using a scientific calculator or CAS (Computer Algebra System) software.