The air pressure in the cabin of a passenger plane is modelled by the equation: P(x) = 3cos(x/2) - sin(x/2) where x is the altitude. Express P(x) in the form Rcos(x/2 +z) where z is acute and in degrees and then find the maximum pressure
This is a classic example of a trigonometry problem using the angle addition formulae. Recall that cos(A+B) = cos(A)cos(B) - sin(A)sin(B). Express R(x/2 -z) in similar form: Rcos(x/2 +z) = Rcos(x/2)cos(z) - Rsin(x/2)sin(z) = 3cos(x/2) - sin(x/2) = P(x) we can now equate coefficients: (1) Rcos(z) = 3 and (2) Rsin(z) = 1 , squaring both equations (1) and (2) and adding them together gives R2(cos2(z) + sin2(z)) = 10 recall that cos2(z) + sin2(z) = 1 therefore R = sqrt(10), dividing equations (1) and (2) gives tan(z) = 1/3 gives the acute angle z = 18.4 degrees. So P(x) = 10-1/2cos( x/2+ 18.4) , the maximum value of P(x) will occur when cos = 1 (maximum value a cosine function can take) therefore R = 10-1/2 is the maximum pressure in the cabin.
