Trig: Show cos(x) + cos(x + θ) = A cos(x + φ), where φ = θ/2 and A = 2 cos(θ/2)

[sherifkaleem]

Trig: Show cos(x) + cos(x + θ) = A cos(x + φ), where φ = θ/2 and A = 2 cos(θ/2)

doing my trig homework, ive done the first part but i have no clue how to do b and c. Please help

(a) State, without proof, the compound angle formula for cos(α + β) and cos(α − β).[2 marks]
(b) Let θ be a fixed real number with 0 ≤ θ < 2π. Show that, for all real x, cos(x) +cos(x + θ) = A cos(x + φ) where φ = θ/2 and A = 2 cos(θ/2) (Hint: use part (a)above). [9 marks]
(c) Determine θ if A = 1/4 and if A = −1. [9 marks]

The physical interpretation of the result in part (b) above (when A ≥ 0) is that thebasic wave cos(x) and the phase-shifted wave cos(x + θ) interfere with each other toproduce another wave with amplitude A and phase shift φ. If A is small then we havedestructive interference. This phenomenon is exploited in noise-cancelling headphonesas used in aircraft for example.

 

[Dr.Peterson]

doing my trig homework, ive done the first part but i have no clue how to do b and c. Please help

(a) State, without proof, the compound angle formula for cos(α + β) and cos(α − β).[2 marks]
(b) Let θ be a fixed real number with 0 ≤ θ < 2π. Show that, for all real x, cos(x) +cos(x + θ) = A cos(x + φ) where φ = θ/2 and A = 2 cos(θ/2) (Hint: use part (a)above). [9 marks]
(c) Determine θ if A = 1/4 and if A = −1. [9 marks]

Well, they gave you a clue: use the compound angle formulas (angle sum formulas). And since a half angle is involved, you can expect also to use either the half-angle formulas (if you have learned them), or the double-angle formulas, which are obtained directly from those in (a).

I would start by writing out the equation with A and B replaced by their expressions, and then apply the angle sum formulas wherever applicable.

Please show your work as far as you get, so we can see what help you need.