Phasors and their corresponding sinusoids are interrelated, offering unique insights into the behavior of alternating current (AC) circuits. One way to understand this relationship is through the operations of differentiation and integration in both the time and phasor domains.

When the derivative of a sinusoid is taken in the time domain, it transforms into its corresponding phasor multiplied by j-omega (jω) in the phasor domain, where j is the imaginary unit, and ω is the angular frequency. Conversely, when a sinusoid is integrated in the time domain, it translates into its corresponding phasor divided by j-omega in the phasor domain. These transformations provide a means to find steady-state solutions for the sinusoid without knowing the initial variable values.

Next, consider two phasors, each represented in rectangular and polar forms. To add or subtract these two phasors, their rectangular forms are used (which express the phasor as a complex number with real and imaginary parts). The real part of the resultant phasor is the sum (for addition) or difference (for subtraction) of the real parts of the two original phasors, and its imaginary part is the sum or difference of the imaginary parts of the individual phasors.

When multiplying or dividing any two phasors, their polar forms are used (expressing the phasor as a magnitude and an angle). The magnitude of the resultant phasor is the product (for multiplication) or quotient (for division) of the magnitudes of the two original phasors, and the angle of the resultant phasor is the sum or difference of the angles of the individual phasors.

Lastly, the complex conjugate of a phasor - which is obtained by changing the sign of its imaginary part - can be expressed in both rectangular and polar forms. This operation is crucial in many applications, including the computation of power in AC circuits.

In conclusion, phasors serve as a powerful mathematical tool in the study of AC circuits, simplifying analysis and solving problems that would be significantly more complex in the time domain.