Final answer:
To sketch two sine waves with one wave lagging by 90 degrees, plot wave A using the equation y1(x) = sin(x) and wave B as y2(x) = cos(x), demonstrating wave B reaching its peaks a quarter cycle after wave A.
Step-by-step explanation:
To sketch two sine waves where sine wave A is the reference and sine wave B lags by 90 degrees, we begin with the standard sine wave equation y = A sin(wt + φ), where A is the amplitude, w is the angular frequency, and φ is the phase shift. A 90-degree phase shift corresponds to a quarter of a full cycle since a full cycle is 360 degrees. In radians, this is π/2 radians since there are 2π radians in a full cycle.
Wave A can be represented as y1(x) = sin(x). For wave B, which lags by 90 degrees or π/2 radians, the equation is y2(x) = sin(x - π/2), which is equivalent to y2(x) = cos(x) due to the trigonometric identity sin(θ - π/2) = cos(θ).
When plotting these two waves on the same graph, they will intersect periodically, but wave B will reach its maximum and minimum values a quarter cycle after wave A does. This illustrates the characteristic relationship between sine and cosine functions where the cosine function is essentially a sine wave shifted to the left by π/2 radians.