To use the phasor transform and its inverse with sinusoidal functions. we are developing a technique for analyzing the steady-state response of a circuit whose source is sinusoidal. performing this analysis in the time domain, even for a simple circuit with a sinusoidal voltage source, a resistor, and inductor in series, results in a differential equation whose solution is difficult to calculate. as often happens in engineering problems, we can transform the problem from the time domain to a different mathematical "space," in this case the frequency domain. in the frequency domain, we can ana much easier to solve. here we introduce the method for transforming from the time domain to the frequency domain. we always start in the time domain by representing any sinusoidal functions using the cosine (not the sine) function. the phasor that corresponds to a given cosine function is a complex number that carries the amplitude and phase angle of the cosine, but not its frequency. for example, consider a circuit with a sinusoidal current source described by the following time-domain equation: i(t) = im cos(wt+0). we can perform the phasor-transform operation on this sinusoidal current to calculate the phasor current as follows:
i= p{i(t)} = p{im cos(wt + 0)} = im 20.

the phasor-transform operation takes a time-domain function described using the cosine function and represents it as a phasor, which is a complex number that in polar form has the magnitude and phase angle of the cosine function. phasors exist in the frequency domain where the frequency of the phasor is a constant and is not explicitly represented by the value of the phasor. we will use the phasor transform operation to transform a circuit with a sinusoidal source in the time domain to a circuit with a phasor source in the frequency domain. we will then use familiar circuit analysis to solve the circuit in the frequency domain; our solution will consist of phasor voltages and currents. but we ultimately want the steady-state time-domain solution, so we will need to transform the phasor voltages and currents back to the appropriate sinusoidal time-domain voltages and currents. we do this by using the inverse-phasor-transform operation on the phasor as follows:

i(t) = p-{i} = p-! {imzp} = im cos(wt +0).
note that we need to know the frequency of the time-domain source to use the inverse-phasor transform operation. that's because the phasor contains only the magnitude and phase angle of the time-domain cosine waveform, but not its frequency. part d - use phasors to find the value for a sum of sinusoids suppose you want to find the sum of two sinusoidal voltages, given as follows: vi(t) = v1 cos(wt+01) and v2 (t) = v2 cos(wt+ if you stay in the time domain, you will have to use trigonometric identities to perform the addition. but if you transform to the frequency domain, you can simply add the phasors v1 and v2 as complex numbers using your calculator. your answer will be a phasor, so you will need to inverse phasor-transform it to get the answer in the time domain. this is an example of a problem that is easier to solve in the frequency domain than in the time domain. use phasor techniques to find an expression for v(t) expressed as a single cosine function, where u(t) = [100 cos(300t + 45°) + 500 cos(300t – 60°)] v. enter your expression using the cosine function. round real numbers using two digits after the decimal point. any angles used should be in degrees. ivo asp u vec vec o ? (t) =