One of my favorite results to share when teaching trigonometry is that \( \sin{x} + \cos{x} \) is secretly a sine function!
\[\sin{x} + \cos{x} \]
\[ \sqrt2 (\frac{1}{\sqrt2}\sin{x}+\frac{1}{\sqrt2}\cos{x})\]
\[ \sqrt2 (\cos{\frac{\pi}{4}\sin{x}+\sin{\frac{\pi}{4}\cos{x})}}\]
\[ \sqrt2 \sin{(x+\frac{\pi}{4})}\]
The process is completely generalizable to \( A\sin{x} + B\cos{x}\)! More here: https://mrhonner.com/archives/18448
