Correct Answer - Option 2 :
\(\frac{1}{{\sqrt 2 }}, + 45^\circ\)
Concept:
For an LTI system, the sinusoidal input produces a sinusoidal output of the same frequency but different amplitude and phase. The amplitude and phase depend upon the transfer function of the system.
For example:
Let, r(t) = cos(ωt) then
y(t) = a cos(ωt + ϕ)
where a = |H(jω)|, ϕ = ∠ H(jω)
\(H(jω)=\frac{Y(jω)}{X(jω)}\)
Calculation:
\(H(jω)=\frac{jω}{jω+2}\)
r(t) = cos (2t)
ω = 2
\(H(jω)=\frac{j2}{j2+2}\)
\(|H(jω)|=\frac{{1}}{{\sqrt 2}}\)
∠ H(jω) = 90° - 45°
= +45°
\(y(t)=\frac{1}{\sqrt 2}~cos (2t + 45^{\circ})\)