1. **State the problem:** Solve the equation $$\sqrt{10}\sin(\theta - 71.56) = -2$$ for $\theta$. 2. **Recall the range of sine function:** The sine function $\sin(x)$ always satisfies $$-1 \leq \sin(x) \leq 1$$ for all real $x$. 3. **Analyze the equation:** The left side is $\sqrt{10} \sin(\theta - 71.56)$. Since $\sqrt{10} \approx 3.162$, the maximum and minimum values of the left side are $$3.162 \times 1 = 3.162$$ and $$3.162 \times (-1) = -3.162$$ respectively. 4. **Check if the right side value is possible:** The equation requires $$\sqrt{10} \sin(\theta - 71.56) = -2$$. Since $-2$ lies between $-3.162$ and $3.162$, the equation is solvable. 5. **Isolate the sine term:** $$\sin(\theta - 71.56) = \frac{-2}{\sqrt{10}}$$ Calculate the right side: $$\frac{-2}{3.162} \approx -0.6325$$ 6. **Find the general solution for $\sin x = -0.6325$:** Let $$x = \theta - 71.56$$. The solutions for $\sin x = a$ are: $$x = \arcsin(a) + 360k$$ or $$x = 180 - \arcsin(a) + 360k$$ for any integer $k$. Since $a = -0.6325$, compute: $$\arcsin(-0.6325) \approx -39.3^\circ$$ 7. **Write the solutions:** $$\theta - 71.56 = -39.3 + 360k$$ or $$\theta - 71.56 = 180 - (-39.3) + 360k = 219.3 + 360k$$ 8. **Solve for $\theta$:** $$\theta = 71.56 - 39.3 + 360k = 32.26 + 360k$$ $$\theta = 71.56 + 219.3 + 360k = 290.86 + 360k$$ where $k$ is any integer. **Final answer:** $$\boxed{\theta = 32.26^\circ + 360^\circ k \quad \text{or} \quad \theta = 290.86^\circ + 360^\circ k, \quad k \in \mathbb{Z}}$$