Use the trig identity:
sinA+sinB=2sin(½(A+B))cos(½(A-B))
When B=A+C, this becomes 2sin(A+½C)cos(½C).
A=π(4.30x-1115t) and C=-0.250π=-π/4.
We can incorporate the common amplitude 4.75 into this to give us:
9.5sin(A+½C)cos(π/8) because cos(-π/8)=cos(π/8).
(a) The new amplitude is 9.5cos(π/8)=8.7769 approx.
(b) Frequency=F=1/Period=1/T, where T can be found by examining the sine argument.
So, π(4.30x-1115t-0.125)=2πn where n is an integer makes sine=0.
That is, 4.30x-1115t-0.125=2n. Initially at (x₀,t₀), n=0 and t₀=(4.30x₀-0.125)/1115.
After the first cycle, t₁=(4.30x₁-2.125)/1115. T=t₁-t₀ and F=1/(t₁-t₀).
F=1115/(4.30(x₁-x₀)-2). If x₀=0, then we can write x₀ simply as x,
F=1115/(4.30x-2).