I enjoy finding calculations that I would consider reasonable for my students to perform that their calculators cannot.

Using a double-angle identity and the pythagorean identity, it’s pretty straightforward to show that \(\sin\left(2\sin^{-1}\left(-\frac{4}{5}\right)\right) = -\frac{24}{25}.\) However, my TI-89 returns an unhelpful \(-\sin(2\sin^{-1}(4/5))\) instead.

Amusingly, the calculator does know that \(2\sin\left(\sin^{-1}\left(-\frac{4}{5}\right)\right)\cos\left(\sin^{-1}\left(-\frac{4}{5}\right)\right) = -\frac{24}{25}.\)

(My calculator is running the 2005 version of the software. It’s possible this has been fixed since then.)