MHB Need help understanding phase shift in trigonometric curves

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The discussion centers on determining the value of α that aligns the trigonometric curves y=asin(2π/λ(x+α)) and z=asin(2π/λx) in phase. The original poster's answer booklet states α=1−λx+nλ, while they consistently arrive at α=nλ, where n is an integer. Participants confirm that both curves have a period of λ, indicating they will be in phase when α equals kλ, with k being any integer. Clarification on the correct interpretation of the equations and phase alignment is sought. The conversation emphasizes the importance of understanding phase shifts in sinusoidal functions.
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At what value of α is the curve y=asin2π/λ (x+α) in phase with z=asin2π/λ(x)?

My answer booklet says α=1−λx+nλ, but I keep getting α=nλ, where n=0,1,2...
I have no clue how to get to the answer shown in the mark scheme. Any insight would be much appreciated!
 
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Guessing at your syntax ... are the following interpretations correct?

$y = A\sin\left[\dfrac{2\pi}{\lambda} \cdot (x + \alpha)\right]$

$z = A\sin\left(\dfrac{2\pi}{\lambda} \cdot x \right)$

If so, then the period of both sinusoids is $\lambda$, hence I agree that the two will be in phase for $\alpha = k \lambda \, , \, k \in \mathbb{Z}$
 
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