The graph shown in the image resembles a cosine function that has been shifted. Let’s analyze it.

• The function starts at its maximum value at $\theta = 0$ and continues in a periodic manner, resembling the cosine graph.
• A normal cosine function starts at $y = 1$ at $\theta = 0$, but this graph seems horizontally shifted by $\frac{\pi}{2}$ to the right.
• The general equation for a cosine function that is shifted to the right by $\frac{\pi}{2}$ is $y = \cos\left(\theta - \frac{\pi}{2}\right)$.

Now, let's match the given options with this observation:

1. The graph of $y = \sin(\theta)$ shifted to the left by $\frac{\pi}{2}$ units – This is not correct because it's a cosine function, not a sine.
2. The graph of $y = \sin(\theta)$ shifted to the right by $\frac{\pi}{2}$ units – This is incorrect, since it is not a sine function.
3. $y = \cos\left(\theta - \frac{\pi}{2}\right)$ – This is the correct answer, as this describes a cosine graph shifted to the right by $\frac{\pi}{2}$.
4. $y = \sin(\theta)$ shifted to the right by $\pi$ units – Incorrect, as this is a sine function shift, not cosine.

The correct option is: $y = \cos\left(\theta - \frac{\pi}{2}\right)$.

Do you want further details or have any questions?

Here are 5 related questions to expand your understanding:

1. How does a horizontal shift affect the equation of sine and cosine functions?
2. What is the difference between the graphs of sine and cosine functions?
3. How would you describe a vertical shift in a trigonometric function?
4. What happens to the period of the function when the graph is stretched horizontally?
5. How can we determine if the shift is to the left or right?

Tip: Always remember, a horizontal shift in trigonometric functions is indicated inside the function’s parentheses, with $+$ for shifts to the left and $-$ for shifts to the right.