Final Answer:
x₁[n] = 2cos(0.2πn + π/4), which represents the sampled signal derived from x(t) = 2cos(10πt + π/4) at a sampling frequency of 10Hz.
Step-by-step explanation:
To generate x₁[n] from x(t), we need to sample x(t) at a frequency of fₛ₁ = 10Hz. The formula for sampling a continuous signal x(t) at a sampling frequency fₛ is x[n] = x(nT), where T = 1/fₛ is the sampling period. In this case, x(t) = 2cos(10πt + π/4) and fₛ₁ = 10Hz. The sampling period T₁ = 1/10 = 0.1 seconds.
Using the formula x₁[n] = x(nT₁), we substitute t = nT₁ into x(t) = 2cos(10πt + π/4) to get x₁[n]. Substituting T₁ = 0.1 seconds into x(t), we get x₁[n] = 2cos(10π * 0.1n + π/4). Simplifying further, x₁[n] = 2cos(πn/5 + π/4).
Therefore, the signal x₁[n] sampled from x(t) at a sampling frequency of 10Hz is x₁[n] = 2cos(0.2πn + π/4). This discrete signal represents the sampled version of the original continuous signal x(t) at the specified sampling frequency, preserving its frequency content at a discrete level suitable for digital processing or analysis.