If tanh x=4/5, find the values of the other hyperbolic functions at x.
sinh x=
cosh x=
coth x=
sech x=
csch x=
I got sinh x=4/3
cosh x=5/3
coth x=5/4
sech x=3/5
csch x=3/4
remember your trigh identities:
tanh = sinh/cosh
sech^2 = 1-tanh^2
and so on.
To find the values of the other hyperbolic functions at x, we can use the identities:
sinh(x) = (e^x - e^(-x)) / 2
cosh(x) = (e^x + e^(-x)) / 2
tanh(x) = sinh(x) / cosh(x)
coth(x) = 1 / tanh(x)
sech(x) = 1 / cosh(x)
csch(x) = 1 / sinh(x)
Given that tanh(x) = 4/5, let's find the values of the other hyperbolic functions:
1. sinh(x):
Using the identity sinh(x) = (e^x - e^(-x)) / 2, we can rearrange the equation as:
e^x - e^(-x) = 2 * sinh(x)
Plugging in the given value of tanh(x) = 4/5, we can use the identity tanh(x) = sinh(x) / cosh(x) to simplify the equation:
sinh(x) = tanh(x) * cosh(x) = (4/5) * cosh(x)
Using the identity cosh(x) = 1 / sqrt(1 - sinh^2(x)), we can further simplify the equation:
sinh(x) = (4/5) * (1 / sqrt(1 - sinh^2(x)))
Now, we can solve this equation for sinh(x) by substituting the given value of tanh(x):
sinh(x) = (4/5) * (1 / sqrt(1 - (4/5)^2))
sinh(x) = (4/5) * (1 / sqrt(1 - 16/25))
sinh(x) = (4/5) * (1 / sqrt(9/25))
sinh(x) = (4/5) * (1 / (3/5))
sinh(x) = (4/5) * (5/3)
sinh(x) = 4/3
So, sinh(x) = 4/3.
2. cosh(x):
Using the identity cosh(x) = (e^x + e^(-x)) / 2, we can rearrange the equation as:
e^x + e^(-x) = 2 * cosh(x)
Plugging in the given value of tanh(x) = 4/5, we can use the identity tanh(x) = sinh(x) / cosh(x) to simplify the equation:
cosh(x) = sinh(x) / tanh(x) = (4/3) / (4/5) = 5/3
So, cosh(x) = 5/3.
3. coth(x):
Using the identity coth(x) = 1 / tanh(x), we can substitute the given value of tanh(x):
coth(x) = 1 / (4/5) = 5/4
So, coth(x) = 5/4.
4. sech(x):
Using the identity sech(x) = 1 / cosh(x), we can substitute the previously found value of cosh(x):
sech(x) = 1 / (5/3) = 3/5
So, sech(x) = 3/5.
5. csch(x):
Using the identity csch(x) = 1 / sinh(x), we can substitute the previously found value of sinh(x):
csch(x) = 1 / (4/3) = 3/4
So, csch(x) = 3/4.
Therefore, the values of the other hyperbolic functions at x are:
sinh(x) = 4/3
cosh(x) = 5/3
coth(x) = 5/4
sech(x) = 3/5
csch(x) = 3/4.
To find the values of the other hyperbolic functions at x when given that tanh x = 4/5, we can use the relationships between the hyperbolic functions.
1. sinh x:
We know that sinh x = tanh x * cosh x. Plugging the given value of tanh x = 4/5 into the formula, we get:
sinh x = (4/5) * cosh x
2. cosh x:
We know that cosh x = 1 / sech x. Rearranging the formula, we can solve for cosh x:
cosh x = 1 / sech x = 1 / √(1 - tanh^2x)
Plugging in the given value of tanh x = 4/5, we have:
cosh x = 1 / √(1 - (4/5)^2)
3. coth x:
We know that coth x = 1 / tanh x. Plugging in the given value of tanh x = 4/5, we have:
coth x = 1 / (4/5)
4. sech x:
We know that sech x = 1 / cosh x. Plugging in the value of cosh x from step 2, we have:
sech x = 1 / cosh x = 1 / (1 / √(1 - (4/5)^2))
5. csch x:
We know that csch x = 1 / sinh x. Plugging in the value of sinh x from step 1, we have:
csch x = 1 / sinh x = 1 / ((4/5) * cosh x)
Now let's calculate the numerical values of these hyperbolic functions:
1. sinh x = (4/5) * cosh x
2. cosh x = 1 / √(1 - (4/5)^2)
3. coth x = 1 / (4/5)
4. sech x = 1 / (1 / √(1 - (4/5)^2))
5. csch x = 1 / ((4/5) * cosh x)
Evaluate these equations to get the precise numerical values of the hyperbolic functions at x.