1. The problem asks us to estimate $\sin\left(\frac{1}{2}\right)$ using linear approximation. 2. Linear approximation uses the formula $f(x) \approx f(a) + f'(a)(x - a)$ near a point $a$ where values are easier to compute. 3. We choose $a=0$ because $\sin(0)=0$ and derivatives of sine at 0 are straightforward. 4. Calculate $f(a) = \sin(0) = 0$. 5. Calculate $f'(x) = \cos(x)$, so $f'(0) = \cos(0) = 1$. 6. Apply the linear approximation formula: $$\sin\left(\frac{1}{2}\right) \approx 0 + 1 \times \left(\frac{1}{2} - 0\right) = \frac{1}{2} = 0.5$$ 7. Thus, the linear approximation estimate for $\sin\left(\frac{1}{2}\right)$ is $0.5$. 8. The actual value is about $0.4794$, so the approximation is close. Final answer: $0.5000$