1. **State the problem:** Solve the equation $$\sqrt{10}\sin(\theta - 71.56) = -2$$ for $$\theta$$ in the interval $$0 \leq \theta \leq 360$$ degrees. 2. **Analyze the equation:** The equation involves the sine function multiplied by $$\sqrt{10}$$. We want to isolate $$\sin(\theta - 71.56)$$. 3. **Isolate the sine term:** $$\sin(\theta - 71.56) = \frac{-2}{\sqrt{10}}$$ 4. **Evaluate the right side:** $$\frac{-2}{\sqrt{10}} = -\frac{2}{\sqrt{10}} = -\frac{2\sqrt{10}}{10} = -\frac{\sqrt{10}}{5} \approx -0.6325$$ 5. **Check the range of sine:** Since sine values range from -1 to 1, $$-0.6325$$ is valid. 6. **Find the reference angle:** $$\alpha = \arcsin(0.6325) \approx 39.23^\circ$$ 7. **Find general solutions for $$\theta - 71.56$$:** Since sine is negative, solutions are in the third and fourth quadrants: - Third quadrant: $$180^\circ + \alpha = 180 + 39.23 = 219.23^\circ$$ - Fourth quadrant: $$360^\circ - \alpha = 360 - 39.23 = 320.77^\circ$$ 8. **Solve for $$\theta$$:** $$\theta = 219.23 + 71.56 = 290.79^\circ$$ $$\theta = 320.77 + 71.56 = 392.33^\circ$$ 9. **Check interval $$0 \leq \theta \leq 360$$:** Only $$290.79^\circ$$ is within the interval. **Final answer:** $$\theta \approx 290.79^\circ$$