Question

which is not a correct description of the graph below?

( y = sin \theta ) shifted to the left by ( pi ) units

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

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

( y = sin ( \theta + pi ) )

Explanation:

Step1: Recall the transformation rules for sine and cosine functions

The general form of a horizontal shift for a function \(y = f(\theta)\) is \(y=f(\theta + h)\), where \(h>0\) shifts the graph to the left by \(h\) units and \(h < 0\) shifts the graph to the right by \(|h|\) units. Also, we know the identity \(\sin(\theta+\pi)=-\sin\theta\) and \(\sin(\theta+\frac{\pi}{2})=\cos\theta\).

Step2: Analyze each option

• Option 1: \(y = \sin\theta\) shifted to the left by \(\pi\) units

Using the transformation rule \(y = f(\theta+h)\), for \(f(\theta)=\sin\theta\) and \(h = \pi\), we get \(y=\sin(\theta+\pi)\). Since \(\sin(\theta+\pi)=-\sin\theta\), this is a correct description.

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

Using the transformation rule \(y = f(\theta + h)\) with \(f(\theta)=\sin\theta\) and \(h=\frac{\pi}{2}\), we get \(y=\sin(\theta+\frac{\pi}{2})\). But \(\sin(\theta+\frac{\pi}{2})=\cos\theta\), and the given graph is \(y =-\sin\theta\) (not \(y=\cos\theta\) shifted left by \(\frac{\pi}{2}\) in the way described).

• Option 3: The graph of \(y = \cos\theta\) shifted to the left by \(\frac{\pi}{2}\) units

Using the transformation rule \(y=f(\theta + h)\) with \(f(\theta)=\cos\theta\) and \(h=\frac{\pi}{2}\), we get \(y=\cos(\theta+\frac{\pi}{2})\). Since \(\cos(\theta+\frac{\pi}{2})=-\sin\theta\), this is a correct description.

• Option 4: \(y=\sin(\theta+\pi)\)

We know from the trigonometric identity \(\sin(A + B)=\sin A\cos B+\cos A\sin B\). For \(A=\theta\) and \(B = \pi\), \(\sin(\theta+\pi)=\sin\theta\cos\pi+\cos\theta\sin\pi=-\sin\theta\), which is the equation of the given graph.

Answer:

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