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Inverse hyperbolic sine
The inverse hyperbolic sine can be expressed in terms of logarithms by
Explanation
For the hyperbolic sine we have
so
Multiplying by ey gives
This is a quadratic equation as
Solving by the quadratic formula, we get
Because ey > 0 and (as
), the minus sign is inadmissible and we have
On both sides we take the logarithm
so that
Example 1
You can see that sinh−1(0) = 0, because
sinh−1(0) = ln (0 + √1) = ln 1 = 0