Sinh x + 2 = cosh x
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To solve the equation sinh x + 2 = cosh x, we can use the definition and properties of hyperbolic functions.
First, let's recall the definitions of sinh(x) and cosh(x):
sinh(x) = (e^x - e^(-x))/2
cosh(x) = (e^x + e^(-x))/2
Now, let's substitute these definitions into the equation:
(e^x - e^(-x))/2 + 2 = (e^x + e^(-x))/2
Now, we can multiply both sides of the equation by 2 to get rid of the denominators:
e^x - e^(-x) + 4 = e^x + e^(-x)
Next, let's move e^(-x) terms to one side and e^x terms to another side:
e^x - e^x - e^(-x) - e^(-x) = -4
This simplifies to:
-2e^(-x) = -4
Now, divide both sides of the equation by -2:
e^(-x) = 2
Taking the natural logarithm of both sides:
ln(e^(-x)) = ln(2)
-x = ln(2)
Finally, let's multiply both sides by -1:
x = -ln(2)
Therefore, the solution for the equation sinh(x) + 2 = cosh(x) is x = -ln(2).
sinh^2x + 4sinhx + 4 = cosh^2x = 1+sinh^2x
4sinhx + 4 = 1
sinhx = -3/4
x = -ln2