Question

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which is a correct description of the graph below?
graph of a wave passing through the origin, with key points at -2π, -π, 0, π, 2π
options:

• the graph of ( y = sin \theta ) shifted to the right by ( \frac{pi}{2} ) units
• ( y = sin \theta ) shifted to the right by ( pi ) units
• ( y = cosleft( \theta - \frac{pi}{2}

ight) )

• the graph of ( y = sin \theta ) shifted to the left by ( \frac{pi}{2} ) units

Explanation:

Step1: Recall Sine and Cosine Graphs

The standard graph of \( y = \sin\theta \) passes through the origin \((0,0)\), has a peak at \( \frac{\pi}{2} \), and a trough at \( \frac{3\pi}{2} \). The standard \( y = \cos\theta \) has a maximum at \( \theta = 0 \). The given graph passes through the origin, has a trough at \( -\frac{\pi}{2} \) and \( \frac{3\pi}{2} \), etc. Let's analyze each option.

Step2: Analyze Option 1 (Right by \( \frac{\pi}{2} \))

Shifting \( y = \sin\theta \) right by \( \frac{\pi}{2} \) gives \( y=\sin(\theta - \frac{\pi}{2}) \). Using the identity \( \sin(A - B)=\cos B\sin A-\sin B\cos A \), \( \sin(\theta - \frac{\pi}{2})=-\cos\theta \), which does not match the graph.

Step3: Analyze Option 2 (Right by \( \pi \))

Shifting \( y = \sin\theta \) right by \( \pi \) gives \( y=\sin(\theta - \pi) \). Using the identity \( \sin(\theta - \pi)=-\sin\theta \). The graph of \( y = -\sin\theta \) is a reflection of \( y = \sin\theta \) over the \( x \)-axis. Let's check the graph: at \( \theta = 0 \), \( y = 0 \); at \( \theta=\frac{\pi}{2} \), \( y=-1 \); at \( \theta = \pi \), \( y = 0 \); at \( \theta=\frac{3\pi}{2} \), \( y = 1 \), which matches the given graph (since the given graph has a trough at \( \frac{\pi}{2} \) and peak at \( \frac{3\pi}{2} \), similar to \( y = -\sin\theta \), which is \( y=\sin(\theta - \pi) \)).

Step4: Analyze Option 3 (\( y=\cos(\theta - \frac{\pi}{2}) \))

Using the identity \( \cos(\theta - \frac{\pi}{2})=\sin\theta \) (since \( \cos(A - B)=\cos A\cos B+\sin A\sin B \), so \( \cos(\theta - \frac{\pi}{2})=\cos\theta\cos\frac{\pi}{2}+\sin\theta\sin\frac{\pi}{2}=\sin\theta \)), which is the original sine graph, not matching.

Step5: Analyze Option 4 (Left by \( \frac{\pi}{2} \))

Shifting \( y = \sin\theta \) left by \( \frac{\pi}{2} \) gives \( y=\sin(\theta+\frac{\pi}{2})=\cos\theta \), which has a maximum at \( \theta = 0 \), not matching the given graph.

Answer:

\( y = \sin\theta \) shifted to the right by \( \pi \) units (the second option)