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secant function to the square of the cosecant function (solid black curves) The dashed gray curves are the graphs of the original squared functions Each horizontal division represents p /2 units Each vertical division represents 1 unit The vertical dashed lines are asymptotes of f

= -180

You were introduced to graphs of the basic tangent and cotangent functions in Chap 2 The tangent is the ratio of the sine to the cosine, and the cotangent is the ratio of the cosine to the sine Now we ll see what happens when we alter or combine these functions in a few different ways

Customer/team/event outside the project Missed item/unclear item/new idea/responding to change in work purpose or objectives/responding to change in system or work environment (box on context diagram)/responding to change of available components Identify each item in each source document or component that needs to be changed For each item, use one row There may be multiple items per document For each one, de ne the desired result of the change This section may have some items added at the time of the initial request It will be completed by the project team if the change request is approved It functions as a design statement for the change Required change.

Tangent and cotangent: example 1 In Fig 15-11, the dashed gray curves are superimposed graphs of the tangent and cotangent functions The solid black curves portray the graph of

f (q) = tan q + cot q The graph of f has asymptotes that pass through every point where the independent variable attains an integer multiple of p /2 The period is p The domain is the set of all real numbers except the integer multiples of p /2 The range is the set of reals larger than or equal

Table 11-2 Change request form with eld de nitions (Continued).

f (q )

Figure 15-11

If the open-loop gain were 1.0 at the natural period, undamped oscillation would result: -1.0 go2 = 1.0 _ 1.0 = - Oc This indicates that an infinitesimal change in 1 2 would change c2 enough so that it would never return to equilibrium, and indeed this is the case. To find the gain and phase characteristics of a loop away from its natural period, the vector equation for the inner loop must be solved for various values of input period 701. This entails first finding the gain and phase of the open loop, g,g,. This vector must then be added to the vector 1.0, LO to form the denominator of the equation. Then the closed-loop gain is the quotient of the magnitude of the two vectors, and its phase is the difference between their phase angles.

Graph of the sum of the tangent and cotangent functions (solid black curves) The dashed gray curves are the graphs of the original functions Each division on the horizontal axis represents p /2 units Each vertical division represents 1 unit The vertical dashed lines are asymptotes of f The dependent-variable axis is also an asymptote of f