1. **Problem:** Solve the system $$\begin{cases} x \cos t - y \sin t = a \\ x \sin t + y \cos t = b \end{cases}$$ 2. **Formula and rules:** This is a linear system in $x$ and $y$. We can solve it using substitution or elimination. Here, we use the method of multiplying and adding equations to eliminate one variable. 3. **Step 1:** Multiply the first equation by $\cos t$ and the second by $\sin t$: $$x \cos^2 t - y \sin t \cos t = a \cos t$$ $$x \sin^2 t + y \sin t \cos t = b \sin t$$ 4. **Step 2:** Add these two equations to eliminate $y$: $$x (\cos^2 t + \sin^2 t) = a \cos t + b \sin t$$ 5. **Step 3:** Use the Pythagorean identity $\cos^2 t + \sin^2 t = 1$: $$x = a \cos t + b \sin t$$ 6. **Step 4:** Multiply the first equation by $\sin t$ and the second by $\cos t$: $$x \cos t \sin t - y \sin^2 t = a \sin t$$ $$x \sin t \cos t + y \cos^2 t = b \cos t$$ 7. **Step 5:** Subtract the first from the second to eliminate $x$: $$y (\cos^2 t + \sin^2 t) = b \cos t - a \sin t$$ 8. **Step 6:** Again, use $\cos^2 t + \sin^2 t = 1$: $$y = b \cos t - a \sin t$$ **Final solution:** $$\boxed{\begin{cases} x = a \cos t + b \sin t \\ y = b \cos t - a \sin t \end{cases}}$$ This completes the solution of the system.