1. **State the problem:** We are given the derivative $\frac{dy}{dx} = \sin\left(x + \frac{\pi}{3}\right)$ and the initial condition $y\left(\frac{\pi}{6}\right) = 3$. We need to find the function $y$ in terms of $x$. 2. **Recall the formula:** To find $y$, we integrate the derivative: $$y = \int \sin\left(x + \frac{\pi}{3}\right) dx + C$$ where $C$ is the constant of integration. 3. **Integrate the function:** Using the integral formula $\int \sin(ax + b) dx = -\frac{1}{a} \cos(ax + b) + C$, here $a=1$ and $b= \frac{\pi}{3}$: $$y = -\cos\left(x + \frac{\pi}{3}\right) + C$$ 4. **Use the initial condition to find $C$:** Substitute $x = \frac{\pi}{6}$ and $y = 3$: $$3 = -\cos\left(\frac{\pi}{6} + \frac{\pi}{3}\right) + C$$ Calculate the angle: $$\frac{\pi}{6} + \frac{\pi}{3} = \frac{\pi}{6} + \frac{2\pi}{6} = \frac{3\pi}{6} = \frac{\pi}{2}$$ Since $\cos\left(\frac{\pi}{2}\right) = 0$: $$3 = -0 + C \implies C = 3$$ 5. **Write the final solution:** $$\boxed{y = -\cos\left(x + \frac{\pi}{3}\right) + 3}$$ This is the function $y$ in terms of $x$ satisfying the given conditions.