Question
Show that the function of time $\text{y}=(\sin\omega\text{t}-\cos\omega\text{t})$ represents simple harmonic motion.
$=\sqrt{2}\Big(\sin\omega\text{t}\times\frac{1}{\sqrt{2}}-\cos\omega\text{t}\times\frac{1}{\sqrt{2}}\Big)$
$=\sqrt{2}\Big(\sin\omega\text{t}\cos\frac{\pi}{4}-\cos\omega\text{t}\sin\frac{\pi}{4}\Big)$ $\Big[\because\cos\frac{\pi}{4}=\frac{1}{\sqrt{2}}\text{ and}\sin\frac{\pi}{4}=\frac{1}{\sqrt{2}}\Big]$
$=\sqrt{2}\sin\Big(\omega\text{t}+\frac{\pi}{4}\Big)$ $[\because\sin(\text{A}-\text{B})=\sin\text{A}\cos\text{B}-\cos\text{A}\sin\text{B}]$
Moreover, $\text{y}\Big(\text{t}+\frac{2\pi}{\omega}\Big)=\sqrt{2}\Big(\omega\text{t}+2\pi-\frac{\pi}{4}\Big)=\text{y(t)}$ Hence. it represents simple harmonic motion.Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.