Help with manipulating trigonometric functions

Questions
1

I have an expression of the form:

(20.193*10^(-3)) Cos[120 \[Pi] t] - (30.634*10^(-3)) Sin[120 \[Pi] t] + (-20.193*10^(-3)) E^(-750 t) Cos[250 Sqrt[7] t] + (145.75*10^(-3)) E^(-750 t) Sin[250 Sqrt[7] t]

I want to factor the terms to produce output:

0.03669 Cos[120 \[Pi] t +0.9875] + 0.14714 E^(-750 t) Cos[250 Sqrt[7] t - 1.707]

i.e., express

ACosx + BSinx = C Cos(x+theta)

I have tried TrigFactor but sometimes it produces the result using Cos and sometimes Sin. I want to be able to control which form to use. Any help would be greatly appreciated. Thanks

Testing.nb (9.96 KB)

2

There is sometimes a chance you can control the output using pattern matching replacement like this

yourexpression /. a_*Cos[x_]+b_*Sin[x_]->aFormulaForC*Cos[x+aFormulaForTheta ]

That is going to search through yourexpression trying to find exactly the left hand side of that and if it finds that then replaces that with the right hand side of that. You need to supply the two formulas written in terms of only a,b,x

3

I guess what I am looking for is to implement the identity:

acosx+bsinx=Rcos(x−α)

where

R= \Sqrt[a^2+b^2],  tanα=b/a

Is there any in built command in Mathematica that implements this. I would like to apply to:

(20.193*10^(-3)) Cos[120 \[Pi] t]- (30.634*10^(-3)) Sin[120 \[Pi] t]

Thanks