1. **State the problem:** We need to use linear approximation to estimate $\sin\left(\frac{9}{4}\right)$. 2. **Choose a point for approximation:** Since $\frac{9}{4} = 2.25$, which is near $\pi/2 \approx 1.5708$ or $\pi \approx 3.1416$, we select $a = \frac{\pi}{2}$ for the linear approximation because the sine function and its derivative are well-known there. 3. **Set up the linear approximation formula:** The linear approximation to $f(x) = \sin x$ near $x=a$ is: $$L(x) = f(a) + f'(a)(x - a)$$ where $f'(x) = \cos x$. 4. **Evaluate $f(a)$ and $f'(a)$:** $$f\left(\frac{\pi}{2}\right) = \sin\left(\frac{\pi}{2}\right) = 1$$ $$f'\left(\frac{\pi}{2}\right) = \cos\left(\frac{\pi}{2}\right) = 0$$ 5. **Apply the linear approximation:** $$L\left(\frac{9}{4}\right) = 1 + 0 \times \left(\frac{9}{4} - \frac{\pi}{2}\right) = 1$$ 6. **Interpretation:** The linear approximation at $a=\frac{\pi}{2}$ suggests $\sin\left(\frac{9}{4}\right) \approx 1$. 7. **For improved accuracy, use $a=\pi$, near 3.1416:** $L(x) = \sin(\pi) + \cos(\pi)(x - \pi) = 0 - 1 \times (x - \pi) = \pi - x$ Estimate: $$L\left(\frac{9}{4}\right) = \pi - \frac{9}{4} \approx 3.1416 - 2.25 = 0.8916$$ Rounded to 4 decimal places: $0.8916$. **Final answer:** Using linear approximation near $\pi$, $\sin\left(\frac{9}{4}\right) \approx 0.8916$.