1. **Problem statement:** We are given the differential equation $$\frac{dy}{dx} = \sin x$$ with initial condition $$y(0) = 0$$. We want to estimate $$y(1)$$ using Euler's Method with 4 steps. 2. **Euler's Method formula:** Euler's Method updates the value of $$y$$ using the formula: $$ y_{n+1} = y_n + f(x_n, y_n) \Delta x $$ where $$f(x, y) = \frac{dy}{dx}$$, and $$\Delta x$$ is the step size. 3. **Step size calculation:** Since we want to go from $$x=0$$ to $$x=1$$ in 4 steps, $$ \Delta x = \frac{1 - 0}{4} = 0.25 $$ 4. **Initial values:** $$ x_0 = 0, \quad y_0 = 0 $$ 5. **Calculate each step:** - Step 1: $$x_0 = 0$$ $$f(x_0) = \sin 0 = 0$$ $$\Delta y_1 = f(x_0) \Delta x = 0 \times 0.25 = 0$$ $$y_1 = y_0 + \Delta y_1 = 0 + 0 = 0$$ - Step 2: $$x_1 = 0.25$$ $$f(x_1) = \sin 0.25 \approx 0.2474$$ (given as 0.311 in table but we use exact sine) $$\Delta y_2 = f(x_1) \Delta x = 0.2474 \times 0.25 = 0.06185$$ $$y_2 = y_1 + \Delta y_2 = 0 + 0.06185 = 0.06185$$ - Step 3: $$x_2 = 0.5$$ $$f(x_2) = \sin 0.5 \approx 0.4794$$ (given as 0.1224 in table but we use exact sine) $$\Delta y_3 = f(x_2) \Delta x = 0.4794 \times 0.25 = 0.11985$$ $$y_3 = y_2 + \Delta y_3 = 0.06185 + 0.11985 = 0.1817$$ - Step 4: $$x_3 = 0.75$$ $$f(x_3) = \sin 0.75 \approx 0.6810$$ (given as 0.2585 in table but we use exact sine) $$\Delta y_4 = f(x_3) \Delta x = 0.6810 \times 0.25 = 0.17025$$ $$y_4 = y_3 + \Delta y_4 = 0.1817 + 0.17025 = 0.35195$$ 6. **Estimate at $$x=1$$:** $$ y(1) \approx y_4 = 0.35195 $$ 7. **Actual value:** The exact solution is $$ y = -\cos x + C $$ Using initial condition $$y(0) = 0$$: $$ 0 = -\cos 0 + C \Rightarrow C = 1 $$ So, $$ y = 1 - \cos x $$ At $$x=1$$: $$ Y = 1 - \cos 1 \approx 1 - 0.5403 = 0.4597 $$ **Summary table:** | x | 0.00 | 0.25 | 0.50 | 0.75 | 1.00 | |-----|-------|-------|-------|-------|-------| | y | 0 | 0 | 0.06185 | 0.1817 | 0.35195 | | dy/dx | 0 | 0.2474 | 0.4794 | 0.6810 | 0.8415 | | Δy | | 0.06185 | 0.11985 | 0.17025 | | | Y | 0 | 0.0311 | 0.1224 | 0.2585 | 0.4597 | Note: The table's given dy/dx and Y values differ from exact sine and exact solution; here we used exact sine values for Euler's method. **Final answer:** $$ \boxed{y(1) \approx 0.352} $$