Question

which is not a correct description of the graph below? y = sin(θ + π), y = sin θ shifted to the left by π units, the graph of y = cos θ shifted to the left by π/2 units, the graph of y = sin θ shifted to the left by π/2 units

Explanation:

Step1: Recall trigonometric identities and transformation rules

The graph of $y = f(x + a)$ is the graph of $y=f(x)$ shifted to the left by $a$ units. Also, $\sin(\theta+\frac{\pi}{2})=\cos\theta$. The graph of $y = \sin\theta$ shifted to the left by $\frac{\pi}{2}$ units gives $y=\sin(\theta+\frac{\pi}{2})=\cos\theta$. The graph of $y = \sin\theta$ shifted to the left by $\pi$ units gives $y=\sin(\theta + \pi)=-\sin\theta$.

Step2: Analyze each option

• For $y=\sin(\theta+\pi)$, it is the graph of $y = \sin\theta$ shifted to the left by $\pi$ units.
• For $y=\sin\theta$ shifted to the left by $\pi$ units is $y=\sin(\theta+\pi)$.
• For the graph of $y = \cos\theta$ is the graph of $y=\sin\theta$ shifted to the left by $\frac{\pi}{2}$ units since $\cos\theta=\sin(\theta+\frac{\pi}{2})$.
• The graph of $y=\sin\theta$ shifted to the left by $\frac{\pi}{2}$ units is a correct transformation related to the cosine - sine relationship.

The incorrect description is that $y=\sin\theta$ shifted to the left by $\pi$ units does not match the given graph (the given graph is more related to the $\sin\theta$ to $\cos\theta$ shift).

Answer:

The graph of $y = \sin\theta$ shifted to the left by $\pi$ units.