For anyone who like me is used to seeing the general solution to $y'' = -\beta^2 y$ written in terms of both sine and cosine as $A\cos(\beta t) + B\sin(\beta t)$, it looks like the form on the slide is equivalent by the cosine addition formula:
$a\cos(t\sqrt{\frac{g}{L}} + b) = a[\cos(t\sqrt{\frac{g}{L}})\cos(b) + \sin(t\sqrt{\frac{g}{L}})\sin(b)] = a\cos(b)\cos(t\sqrt{\frac{g}{L}}) + a\sin(b)\sin(t\sqrt{\frac{g}{L}}) = A\cos(t\sqrt{\frac{g}{L}}) + B\sin(t\sqrt{\frac{g}{L}})$
Background in 21260 helps. The fact that there is no close form solution is still bizarre to me.
For anyone who like me is used to seeing the general solution to $y'' = -\beta^2 y$ written in terms of both sine and cosine as $A\cos(\beta t) + B\sin(\beta t)$, it looks like the form on the slide is equivalent by the cosine addition formula:
$a\cos(t\sqrt{\frac{g}{L}} + b) = a[\cos(t\sqrt{\frac{g}{L}})\cos(b) + \sin(t\sqrt{\frac{g}{L}})\sin(b)] = a\cos(b)\cos(t\sqrt{\frac{g}{L}}) + a\sin(b)\sin(t\sqrt{\frac{g}{L}}) = A\cos(t\sqrt{\frac{g}{L}}) + B\sin(t\sqrt{\frac{g}{L}})$
Background in 21260 helps. The fact that there is no close form solution is still bizarre to me.