A particle moves in the x-y plane according to rule $x = a \sin \omega t$ and $y = a \cos \omega t$. The parti

Question

A particle moves in the x-y plane according to rule $x = a \sin \omega t$ and $y = a \cos \omega t$. The particle follows:

Accepted Answer

1. **Identify the Parametric Equations:** The motion of the particle is described by the coordinates: 
   $$ x = a \sin \omega t $$
   $$ y = a \cos \omega t $$
2. **Eliminate the Time Parameter ($t$):** To determine the path (trajectory), we eliminate $t$ by squaring both equations and adding them together.
   $$ x^2 = a^2 \sin^2 \omega t $$
   $$ y^2 = a^2 \cos^2 \omega t $$
   $$ x^2 + y^2 = a^2 (\sin^2 \omega t + \cos^2 \omega t) $$
3. **Apply Trigonometric Identity:** Using $\sin^2 	heta + \cos^2 	heta = 1$:
   $$ x^2 + y^2 = a^2 $$
4. **Conclusion:** The resulting equation represents a **circle** with its center at the origin $(0, 0)$ and a constant radius $a$. Thus, the particle performs uniform circular motion .

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