Verifying Trig Identity

• Mar 9th 2010, 06:05 PM
kaiser0792
Verifying Trig Identity
I have one more that has me stumped. In verifying the following identity,
I came up with a form that was close to the desired form, but ........
Well you know what they say about getting close. I rechecked by algebra and it all looks good. Help!

sinh 3x = 3sinh x + 4 (sinh^3) x

NOTE: In case its not clear the last term is 4 times sinh cubed x, where the sinh function is cubed.
• Mar 9th 2010, 06:20 PM
Prove It
Quote:

Originally Posted by kaiser0792
I have one more that has me stumped. In verifying the following identity,
I came up with a form that was close to the desired form, but ........
Well you know what they say about getting close. I rechecked by algebra and it all looks good. Help!

sinh 3x = 3sinh x + 4 (sinh^3) x

NOTE: In case its not clear the last term is 4 times sinh cubed x, where the sinh function is cubed.

These are not trigonometric functions, they're actually hyperbolic functions.

But never mind...

$\displaystyle \sinh{3x} = \sinh{(2x + x)}$

Use the identity $\displaystyle \sinh{(a + b)} = \sinh{a}\cosh{b} + \cosh{a}\sinh{b}$

$\displaystyle \sinh{(2x + x)} = \sinh{2x}\cosh{x} + \cosh{2x}\sinh{x}$

$\displaystyle = 2\sinh{x}\cosh{x}\cosh{x} + (2\sinh^2{x} + 1)\sinh{x}$

$\displaystyle = 2\sinh{x}\cosh^2{x} + 2\sinh^3{x} + \sinh{x}$

$\displaystyle = 2\sinh{x}(1 + \sinh^2{x}) + 2\sinh^3{x} + \sinh{x}$

$\displaystyle = 2\sinh{x} + 2\sinh^3{x} + 2\sinh^3{x} + \sinh{x}$

$\displaystyle = 3\sinh{x} + 4\sinh^3{x}$.
• Mar 9th 2010, 06:27 PM
Random Variable
$\displaystyle 3 \ \sinh x + 4 \ \sinh^{3} x$

$\displaystyle = 3 \ \frac{e^{x}-e^{-x}}{2} + 4 \ \frac{(e^x-e^{-x})^{3}}{8}$

$\displaystyle = 3 \ \frac{e^{x}-e^{-x}}{2} + \ \frac{e^{3x}-3e^{x}+3^{-x}-e^{-3x}}{2}$

$\displaystyle = \frac{e^{3x}-e^{-3x}}{2} = \sinh 3x$