Question

2. (a) State, without proof, the compound angle formulae for sin(a + β) and sin(α-β). 2 marks (b) Let θ be a fixed real number with 0 < θ < π. Show that, for all real x, sin(z+θ)- sin(z-Asin(z + φ) where φ (π + θ)/2 and A-2 sin(θ/2) (Hint: use part (a) above). 10 marks] (c) Determine φ if A = V2 and if A = V3. [9 marks] The physical interpretation of the result in part (b) above is that the basic wave sin() and the phase-shifted wave sin(r+0) interfere with each other to produce another wave with amplitude A and phase shift φ. If A is small then we have destructive interference. This phenomenon is exploited in noise-cancelling headphones as used in aircraft for example.

Answer 1

A = 9.Sn (012)