Question

Evaluate \[ \log\left(\sinh\theta+\sqrt{\sinh^2\theta+1}\right). \]

• A. \(\cosh\theta\)
• B. \(\sinh^{-1}\theta\)
• C. \(\theta\)
• D. \(\cosh^{-1}\theta\)

Hint:

Remember the standard identities \[ \boxed{\cosh^2x-\sinh^2x=1} \] and \[ \boxed{\sinh x+\cosh x=e^x.} \] These directly simplify logarithmic expressions involving hyperbolic functions.

Answer 1

The Correct Option is C

Step 1: Use the hyperbolic identity. We know that \[ \cosh^2\theta-\sinh^2\theta=1. \] Hence, \[ \sqrt{\sinh^2\theta+1} =\sqrt{\cosh^2\theta} =\cosh\theta, \] since \[ \cosh\theta>0. \]

Step 2: Simplify the logarithm. Therefore, \[ \log\left(\sinh\theta+\sqrt{\sinh^2\theta+1}\right) = \log(\sinh\theta+\cosh\theta). \] Using \[ \sinh\theta+\cosh\theta=e^\theta, \] we get \[ \log(e^\theta)=\theta. \]

Step 3: Write the final answer. Hence, \[ \boxed{\theta}. \] Thus, \[ \boxed{(C)} \] is the correct answer.