Question

the movement of the progress bar may be uneven because questions can be worth more or less (including zero) depending on your answer. which is not a correct description of the graph below? graph of a trigonometric function options: the graph of ( y = sin \theta ) shifted to the left by ( \frac{pi}{2} ) units, ( y = sin \theta ) shifted to the left by ( pi ) units, the graph of ( y = cos \theta ) shifted to the left by ( \frac{pi}{2} ) units, ( y = sin(\theta + pi) )

Explanation:

Step1: Analyze the function \(y = \sin(\theta+\pi)\)

Using the formula \(\sin(A + B)=\sin A\cos B+\cos A\sin B\), for \(y=\sin(\theta+\pi)\), we have \(y=\sin\theta\cos\pi+\cos\theta\sin\pi\). Since \(\cos\pi=- 1\) and \(\sin\pi = 0\), then \(y=-\sin\theta\).

Step2: Analyze the function \(y=\cos\theta\) shifted to the left by \(\frac{\pi}{2}\) units

The rule for horizontal shift of a function \(y = f(x)\) to the left by \(h\) units is \(y=f(x + h)\). For \(y=\cos\theta\) shifted left by \(\frac{\pi}{2}\) units, we get \(y=\cos(\theta+\frac{\pi}{2})\). Using the formula \(\cos(A + B)=\cos A\cos B-\sin A\sin B\), with \(A=\theta\) and \(B = \frac{\pi}{2}\), we have \(y=\cos\theta\cos\frac{\pi}{2}-\sin\theta\sin\frac{\pi}{2}\). Since \(\cos\frac{\pi}{2}=0\) and \(\sin\frac{\pi}{2}=1\), then \(y=-\sin\theta\).

Step3: Analyze the function \(y=\sin\theta\) shifted to the left by \(\pi\) units

Using the horizontal - shift rule \(y = f(x + h)\) (where \(h=\pi\) and \(f(x)=\sin x\)), we get \(y=\sin(\theta+\pi)\) which we already know is \(y =-\sin\theta\).

Step4: Analyze the function \(y=\sin\theta\) shifted to the left by \(\frac{\pi}{2}\) units

Using the horizontal - shift rule \(y=\sin(\theta+\frac{\pi}{2})\). By the formula \(\sin(A + B)=\sin A\cos B+\cos A\sin B\) with \(A=\theta\) and \(B=\frac{\pi}{2}\), we have \(y=\sin\theta\cos\frac{\pi}{2}+\cos\theta\sin\frac{\pi}{2}\). Since \(\cos\frac{\pi}{2}=0\) and \(\sin\frac{\pi}{2}=1\), then \(y=\cos\theta
eq-\sin\theta\)

Answer:

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