1. **State the problem:** Solve for $\theta$ in the equation $$3 \sin\left(\frac{\theta}{2} - 20^\circ\right) = 0.85$$ where $0^\circ \leq \theta \leq 90^\circ$. 2. **Isolate the sine term:** Divide both sides by 3: $$\sin\left(\frac{\theta}{2} - 20^\circ\right) = \frac{0.85}{3} \approx 0.2833$$ 3. **Use the inverse sine function:** Let $$x = \frac{\theta}{2} - 20^\circ$$ Then $$\sin x = 0.2833$$ So, $$x = \sin^{-1}(0.2833)$$ Using a calculator, $$x \approx 16.45^\circ$$ 4. **Consider the sine function's properties:** Since sine is positive in the first and second quadrants, $$x_1 = 16.45^\circ$$ $$x_2 = 180^\circ - 16.45^\circ = 163.55^\circ$$ 5. **Solve for $\theta$:** Recall $$x = \frac{\theta}{2} - 20^\circ$$ So, $$\frac{\theta}{2} = x + 20^\circ$$ For $x_1$: $$\frac{\theta}{2} = 16.45^\circ + 20^\circ = 36.45^\circ$$ $$\theta = 2 \times 36.45^\circ = 72.9^\circ$$ For $x_2$: $$\frac{\theta}{2} = 163.55^\circ + 20^\circ = 183.55^\circ$$ $$\theta = 2 \times 183.55^\circ = 367.1^\circ$$ 6. **Check the domain:** Since $0^\circ \leq \theta \leq 90^\circ$, only $$\theta = 72.9^\circ$$ is valid. **Final answer:** $$\boxed{\theta \approx 72.9^\circ}$$