Response to sinusoidal function

A more useful form is obtained by using the identity  

This type of function is said to be stationary, i.e. its value varies with time, but in a regularly repeating pattern. Stationary conditions should not be confused with the steady state. Comparison of equations 7.71 and 7.89 shows that when t - oo  

If xj/ 0 the output is said to lag the input, and if xj/ 0 it is said to lead the input. The relationship between input and output for a sinusoidal forcing function as t - oo constitutes an important tool in the analysis and design of control systems termed frequency response analysis. Of particular importance are the amplitude ratio (AR) and the phase shift xfr. The AR represents the relationship between the output and the input amplitudes 

The phase shift is the difference in angle measured in degrees between input and output as t -+ oo (Fig. 7.29). 

For the above first-order system, from equations 7.71 and 7.89  

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