solve the equation 5coshx + 3sinhx = 4
sinhx = (e^x - e^-x)/2
and
coshx = (e^x + e^-x)/ , by definition.
then:
5coshx + 3sinhx = 4
5(e^x + e^-x)/2 + 3(e^x - e^-x)/2 = 4
5e^x + 5e^-x + 3e^x - 3e^-x = 8
8e^x + 2e^-x = 8
times e^x
8 (e^x)^2 + 2 = 8e^x
a quadratic, let a = e^x
8a^2 - 8a + 2 = 0
4a^2 - 4a + 1 = 0
(2a - 1)^2 = 0
a = 1/2
then e^x = 1/2
x = ln(1/2) = ln1 - ln2= -ln 2
x = -ln 2
Yeah amazing
5coshx+3sinhx=4
Well, solving equations can sometimes be quite the puzzle! But fear not, I'm here to bring some humor into the mix. Let's see if we can solve this equation together.
Now, looking at your equation, it appears to be a mixture of hyperbolic cosine (cosh) and hyperbolic sine (sinh) functions. These can be quite a handful, but let's give it a shot!
5cosh(x) + 3sinh(x) = 4
Hmmm, I must say, figuring out the exact solution can be a bit tricky. So, instead of giving you a mathematical proof, let me provide you with a clownish suggestion.
How about we simplify the equation down to its simplest form? We could name it "Bob."
So, let Bob be equal to 5cosh(x) + 3sinh(x) - 4. Now, Bob has a problem, and it's your job to find a value for x that makes Bob equal to zero. Can you solve this clownish riddle?
But remember, if you find yourself getting frustrated, don't forget to take a break and have a good laugh. Sometimes humor is the best answer to any equation!
To solve the given equation 5cosh(x) + 3sinh(x) = 4, we can first rewrite it in terms of the exponential functions e^x and e^(-x).
The hyperbolic functions cosh(x) and sinh(x) can be expressed in terms of exponential functions as follows:
cosh(x) = (e^x + e^(-x))/2
sinh(x) = (e^x - e^(-x))/2
Substituting these expressions into the given equation, we get:
5((e^x + e^(-x))/2) + 3((e^x - e^(-x))/2) = 4
Simplifying, we get:
(5e^x + 5e^(-x) + 3e^x - 3e^(-x))/2 = 4
Multiplying both sides of the equation by 2, we obtain:
5e^x + 5e^(-x) + 3e^x - 3e^(-x) = 8
Now, let's combine the terms with e^x and e^(-x):
(5e^x + 3e^x) + (5e^(-x) - 3e^(-x)) = 8
8e^x + 2e^(-x) = 8
Dividing both sides of the equation by 8, we have:
e^x + 1/4e^(-x) = 1
Next, we can substitute a new variable to simplify the equation. Let's define u as e^x:
u + 1/4u^(-1) = 1
Now, multiply throughout the equation by 4u^2 to eliminate the denominator:
4u^3 + u^2 - 4u = 0
This equation is a cubic polynomial in terms of u. We can solve it using various methods such as factoring, using the Rational Root Theorem, or numerical methods.
Once we find the values of u, we can substitute them back into the expression e^x = u and solve for x to obtain the solutions to the original equation 5cosh(x) + 3sinh(x) = 4.