Express 4sinx + 3cosx in the form Rcos(x-a)
From the following identity, cos(a-b) = cosacosb+ sinasinb, we find that 4sinx+3cosx = R(cosxcosa+sinxsina). We now equate the coefficients: 3 = Rcosa and 4=Rsina. Using basic trigonometry, we can make this into a right angled triangle, the side of length 4 being opposite to the angle a, and the side of length 3 being adjacent. The hypotenuse is therefore R, and can be calculated using Pythagoras theorem to give 5. Angle a can also be calculated, as tana = 4/3, hence a = 53.1 degrees. Therefore our answer is 5cos(x-53.1)
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