Question

If $\sinh x = \frac{3}{4}$, then $\cosh 2x =$

• A. $\frac{17}{8}$
• B. $\frac{15}{8}$
• C. $\frac{9}{8}$
• D. $\frac{25}{8}$

Hint:

Hyperbolic trigonometric identities are very similar to standard ones but watch out for sign differences ($\cosh^2 x - \sinh^2 x = 1$ and $\cosh 2x = 1 + 2\sinh^2 x$).

Answer 1

The Correct Option is A

Step 1: Pick a double-angle identity.
For hyperbolic functions, $\cosh 2x = 1 + 2\sinh^2 x$. This uses only $\sinh x$, which we already know.

Step 2: Note the given value.
We are told $\sinh x = \frac34$.

Step 3: Square it.
\[ \sinh^2 x = \left(\frac34\right)^2 = \frac{9}{16} \]

Step 4: Substitute into the identity.
\[ \cosh 2x = 1 + 2 \cdot \frac{9}{16} \]

Step 5: Simplify the product.
\[ 2 \cdot \frac{9}{16} = \frac{9}{8} \]

Step 6: Add.
\[ \cosh 2x = 1 + \frac{9}{8} = \frac{17}{8} \] \[ \boxed{ \cosh 2x = \frac{17}{8} } \]