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harrietstowe
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Homework Statement
I need to prove that tanh-1(x) is equal to (1/2)ln((1+x)/(1-x))
Homework Equations
The Attempt at a Solution
I rewrote it as ln(x+(x2+1)1/2)/ln(x+(x2-1)1/2)
from here I am stuck. Thank you
harrietstowe said:Homework Statement
I need to prove that tanh-1(x) is equal to (1/2)ln((1+x)/(1-x))
Homework Equations
The Attempt at a Solution
I rewrote it as ln(x+(x2+1)1/2)/ln(x+(x2-1)1/2)
from here I am stuck. Thank you
Tanh^-1(x) is the inverse hyperbolic tangent function. It is the inverse of the hyperbolic tangent function, tanh(x), and is used to find the angle whose hyperbolic tangent is x.
Tanh^-1(x) is commonly used in trigonometric proofs involving hyperbolic functions. It allows for the manipulation and simplification of equations involving hyperbolic tangents.
The domain of tanh^-1(x) is (-1,1), as it is the inverse of the hyperbolic tangent function whose range is (-1,1). The range of tanh^-1(x) is (-∞,∞).
To evaluate tanh^-1(x), we use the following formula: tanh^-1(x) = (1/2)ln[(1+x)/(1-x)]. This can also be rewritten as tanh^-1(x) = (1/2)[ln(1+x) - ln(1-x)]. This formula can be derived from the definition of the inverse hyperbolic tangent function.
Some common identities involving tanh^-1(x) include: tanh^-1(-x) = -tanh^-1(x), tanh^-1(0) = 0, and tanh^-1(∞) = ∞. There are also various trigonometric identities that involve tanh^-1(x), such as the double angle formula and the addition/subtraction formula.