1. **State the problem:** Solve the equation $$\cos(x + 30^\circ) - \cos(x + 48^\circ) = 0.2$$ for $$30^\circ \leq x \leq 360^\circ$$. 2. **Use the cosine difference identity:** $$\cos A - \cos B = -2 \sin \left( \frac{A+B}{2} \right) \sin \left( \frac{A-B}{2} \right)$$ where $$A = x + 30^\circ$$ and $$B = x + 48^\circ$$. 3. **Apply the identity:** $$\cos(x + 30^\circ) - \cos(x + 48^\circ) = -2 \sin \left( \frac{(x + 30^\circ) + (x + 48^\circ)}{2} \right) \sin \left( \frac{(x + 30^\circ) - (x + 48^\circ)}{2} \right)$$ Simplify inside the sine functions: $$= -2 \sin \left( \frac{2x + 78^\circ}{2} \right) \sin \left( \frac{-18^\circ}{2} \right) = -2 \sin(x + 39^\circ) \sin(-9^\circ)$$ 4. **Use the odd property of sine:** $$\sin(-9^\circ) = -\sin(9^\circ)$$ So, $$-2 \sin(x + 39^\circ)(-\sin 9^\circ) = 2 \sin(x + 39^\circ) \sin 9^\circ$$ 5. **Rewrite the equation:** $$2 \sin(x + 39^\circ) \sin 9^\circ = 0.2$$ 6. **Divide both sides by $$2 \sin 9^\circ$$:** $$\sin(x + 39^\circ) = \frac{0.2}{2 \sin 9^\circ} = \frac{0.2}{2 \times 0.1564} = \frac{0.2}{0.3128} \approx 0.6397$$ 7. **Solve for $$x + 39^\circ$$:** $$x + 39^\circ = \sin^{-1}(0.6397)$$ The principal value: $$x + 39^\circ \approx 39.8^\circ$$ The sine function is positive in the first and second quadrants, so the second solution is: $$x + 39^\circ = 180^\circ - 39.8^\circ = 140.2^\circ$$ 8. **Find $$x$$ values:** $$x_1 = 39.8^\circ - 39^\circ = 0.8^\circ$$ $$x_2 = 140.2^\circ - 39^\circ = 101.2^\circ$$ 9. **Check domain $$30^\circ \leq x \leq 360^\circ$$:** Only $$x_2 = 101.2^\circ$$ is in the domain. 10. **General solutions:** Add $$360^\circ$$ to find other solutions within the domain: $$x_3 = 0.8^\circ + 360^\circ = 360.8^\circ$$ (exceeds 360°, discard) $$x_4 = 101.2^\circ + 360^\circ = 461.2^\circ$$ (exceeds 360°, discard) **Final solution:** $$x \approx 101.2^\circ$$ within $$30^\circ \leq x \leq 360^\circ$$.