Final answer:
The voltage function v(t) = 10sin(1000πt + 30°) V can be expressed in terms of t and the constant π2 as v(t) = 10cos(1000πt - 60°) V.
Step-by-step explanation:
The given voltage function is v(t) = 10sin(1000πt + 30°) V. To express it in terms of t and the constant π, we can use the cosine function: v(t) = 10cos(1000πt - 60°) V.
The angular frequency (ω) is the coefficient of t in the function: ω = 1000π rad/s.
The frequency (f) is the angular frequency divided by 2π: f = ω/(2π), which is approximately 159.2 Hz. The period (T) is the reciprocal of the frequency: T = 1/f, which is approximately 0.0063 s.
The phase angle (θ) is the constant term in the function: θ = -60°.
The VRMS (root mean square) value can be calculated using the formula: VRMS = Vmax/√2, where Vmax is the peak voltage. In this case, Vmax = 10 V, so VRMS = 10/√2 ≈ 7.07 V.
The power delivered to a resistance (P) can be calculated using the formula: P = (VRMS^2)/R, where R is the resistance. In this case, R = 60 Ω, so P ≈ (7.07^2)/60 ≈ 0.829 W.
The first value after t=0 that v(t) reaches its peak value of 10 V can be found by finding the time when cos(1000πt - 60°) = 1, which is when 1000πt - 60° = 0 or t - 0.06 = 0. Therefore, the first value after t=0 is t = 0.06 s.