Final answer:
The question involves converting phasors into instantaneous time functions for an oscillatory system, possibly an electrical circuit or mechanical system, to find values like current, voltage, or position at specific times. This involves the use of trigonometric functions and concepts such as amplitude, angular frequency, phase shift, and phasor rotation.
Step-by-step explanation:
The question pertains to understanding the relationship between phasors and their corresponding instantaneous time functions. In an oscillatory system, such as an electrical circuit or mechanical system, phasors are a graphical representation of the magnitude and phase of sinusoidal functions over time. The instantaneous values of variables such as current (i(t)), voltage (v(t)), position (x(t)), velocity, and acceleration can be derived from phasors that rotate at a constant angular frequency. For example, phasor diagrams can show how the phasors add up at certain instances, indicating the magnitude and direction of the resultant wave at those points in time.
Given a phasor that rotates through a specific angle, you can determine the instantaneous values for variables like current and voltage using corresponding trigonometric functions. Additionally, one can find the phase shift of a function represented by a phasor, which is an important factor in determining the initial conditions of an oscillatory system.
To determine the instantaneous time functions, you would typically transform the phasor representation into a time domain function by applying Euler's formula or using trigonometric identities. The usual form for a sinusoidal function with a phase shift is x(t) = A cos(ωt + ϕ) or x(t) = A sin(ωt + ϕ), where A is the amplitude, ω is the angular frequency, and ϕ is the phase shift.