Question

If \( \sinh^{-1}(2) + \sinh^{-1}(3) = \alpha \), then \( \sinh\alpha = \) ?

• A. \( 2\sqrt{5} + 3\sqrt{10} \)
• B. \( 2\sqrt{10} + 4\sqrt{5} \)
• C. \( 3\sqrt{10} + 4\sqrt{5} \)
• D. \( 2\sqrt{10} + 3\sqrt{5} \)

Hint:

Remember the identity \( \sinh(x + y) = \sinh x \cosh y + \cosh x \sinh y \) when summing inverse hyperbolic sine expressions.

Answer 1

The Correct Option is D

Step 1: Let \[ x = \sinh^{-1}(2),
y = \sinh^{-1}(3) \Rightarrow \alpha = x + y \] Step 2: Use the identity \[ \sinh(x + y) = \sinh x \cosh y + \cosh x \sinh y \] Step 3: Find \( \sinh x = 2, \sinh y = 3 \) by definition. Now compute: \[ \cosh x = \sqrt{1 + \sinh^2 x} = \sqrt{1 + 4} = \sqrt{5},
\cosh y = \sqrt{1 + \sinh^2 y} = \sqrt{1 + 9} = \sqrt{10} \] Step 4: Plug into identity: \[ \sinh(x + y) = 2 . \sqrt{10} + 3 . \sqrt{5} = 2\sqrt{10} + 3\sqrt{5} \] So, \( \sinh\alpha = \boxed{2\sqrt{10} + 3\sqrt{5}} \)