Topic: Signals and Systems
Find the rms value of signal $x(t)=5~cos(50t+\frac{\pi}{3})$
- A. $\sqrt{5}$
- B. $\sqrt{5\times50t}$
- C. $\sqrt{50+\frac{\pi}{3}}$
- D. $\sqrt{12.5}$
Answer: D. $\sqrt{12.5}$
Explanation: $x(t)=5~cos(50t+\frac{\pi}{3}) = 5\{cos(50t)\cdot cos(\frac{\pi}{3})-sin(50t).sin(\frac{\pi}{3})\}$\n$=5\{cos(50t)\cdot\frac{1}{2}-sin(50t)\frac{\sqrt{3}}{2}\}$\n$x(t)=\frac{5}{2}cos(50t)-\frac{5\sqrt{3}}{2}sin(50t)$\n$x(t)_{rms}=\sqrt{(\frac{5}{2\sqrt{2}})^{2}+(\frac{5\sqrt{3}}{2\sqrt{2}})^{2}}$\n$=\sqrt{(\frac{5}{2\sqrt{2}})^{2}(1+\sqrt{3})} = \sqrt{\frac{25}{8}\times4} = \sqrt{12.5}$