1. **State the problem:** We are given the function $y = 3 \sin\left(\theta - \frac{\pi}{2}\right)$ and want to understand its properties and graph. 2. **Recall the sine function properties:** The general form is $y = A \sin(B(\theta - C))$ where: - $A$ is the amplitude (height of peaks), - $B$ affects the period, - $C$ is the phase shift. 3. **Identify parameters:** Here, $A=3$, $B=1$, and $C=\frac{\pi}{2}$. - Amplitude $= 3$ means the wave oscillates between $-3$ and $3$. - Phase shift $= \frac{\pi}{2}$ to the right. 4. **Use sine phase shift identity:** $$\sin\left(\theta - \frac{\pi}{2}\right) = -\cos(\theta)$$ So, $$y = 3 \sin\left(\theta - \frac{\pi}{2}\right) = 3(-\cos(\theta)) = -3 \cos(\theta)$$ 5. **Interpretation:** The function is equivalent to $y = -3 \cos(\theta)$, a cosine wave flipped vertically and scaled by 3. 6. **Graph features:** - Amplitude: 3 - Period: $2\pi$ (since $B=1$) - Phase shift: $\frac{\pi}{2}$ right - Vertical shift: none 7. **Summary:** The graph oscillates between $-3$ and $3$, starting at $y=0$ when $\theta=0$, moving downward first, consistent with the sine shifted by $\frac{\pi}{2}$. Final answer: $$y = 3 \sin\left(\theta - \frac{\pi}{2}\right) = -3 \cos(\theta)$$