Problem

Compare \(A(x)~=~4\cos(x~-~\pi~/2)\) and \(B(x)~=~4\sin(x)\). Use the graph to state their amplitude, period, midline, and range, interpret A's phase shift, apply the relevant identity, and decide whether the graphs are identical.

Prompt sinusoidal graph on labeled axes showing only the equation or stated geometric source features; requested conclusions are withheld.

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Both have amplitude 4, period 2pi, midline 0, and range [-4, 4]. A is a cosine shifted pi/2 right, and cos(x - pi/2) = sin(x), so they are identical graphs.

A has amplitude 4, period 2pi, midline pi/2, and range [pi/2 - 4, pi/2 + 4]. B has amplitude 4, period 2pi, midline 0, and range [-4, 4]. The subtraction inside A creates a vertical shift, so the graphs are not identical.

A has amplitude 4, period pi, midline 0, and range [-4, 4]. B has amplitude 4, period 2pi, midline 0, and range [-4, 4]. A completes twice as many cycles as B, so the graphs are not identical.

Both have amplitude 4, period 2pi, midline 0, and range [-4, 4]. A is a cosine shifted pi/2 left, and cos(x - pi/2) = -sin(x), so the graphs are reflections across the x-axis.

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Compare \(A(x)~=~4\cos(x~-~\pi~/2)\) and \(B(x)~=~4\sin(x)\). Use the graph to state their amplitude, period, midline, and range, interpret A's phase shift, apply the relevant identity, and decide whether the graphs are identical.

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