1. **State the problem:** We need to find all values of $x$ in the interval $0^\circ \leq x \leq 360^\circ$ such that $\sin x = -0.8176$. 2. **Recall the sine function properties:** The sine function has a range of $[-1,1]$ and is negative in the third and fourth quadrants (i.e., between $180^\circ$ and $360^\circ$). 3. **Find the reference angle:** First, find the angle whose sine is the positive value $0.8176$. $$\theta = \sin^{-1}(0.8176)$$ Using a calculator, $$\theta \approx 55^\circ$$ 4. **Determine the solutions in the given interval:** Since $\sin x$ is negative, the solutions are in the third and fourth quadrants. - Third quadrant solution: $$x = 180^\circ + \theta = 180^\circ + 55^\circ = 235^\circ$$ - Fourth quadrant solution: $$x = 360^\circ - \theta = 360^\circ - 55^\circ = 305^\circ$$ 5. **Final answers:** The values of $x$ are approximately $235^\circ$ and $305^\circ$. **Note:** The user mentioned $55^\circ$ and $235^\circ$; $55^\circ$ corresponds to the positive sine value, but since the sine is negative, the correct solutions are $235^\circ$ and $305^\circ$. Hence, the solutions to $\sin x = -0.8176$ in $0^\circ \leq x \leq 360^\circ$ are: $$x \approx 235^\circ, 305^\circ$$