1. We are asked to solve the equation $\sin(45^\circ - a) - 3\cos(45^\circ + a) + 1 = 0$ for $a$. 2. Recall the angle sum and difference formulas: $$\sin(x - y) = \sin x \cos y - \cos x \sin y$$ $$\cos(x + y) = \cos x \cos y - \sin x \sin y$$ 3. Apply these formulas to the given equation: $$\sin(45^\circ - a) = \sin 45^\circ \cos a - \cos 45^\circ \sin a = \frac{\sqrt{2}}{2} \cos a - \frac{\sqrt{2}}{2} \sin a$$ $$\cos(45^\circ + a) = \cos 45^\circ \cos a - \sin 45^\circ \sin a = \frac{\sqrt{2}}{2} \cos a - \frac{\sqrt{2}}{2} \sin a$$ 4. Substitute these into the original equation: $$\left(\frac{\sqrt{2}}{2} \cos a - \frac{\sqrt{2}}{2} \sin a\right) - 3\left(\frac{\sqrt{2}}{2} \cos a - \frac{\sqrt{2}}{2} \sin a\right) + 1 = 0$$ 5. Distribute the $-3$: $$\frac{\sqrt{2}}{2} \cos a - \frac{\sqrt{2}}{2} \sin a - \frac{3\sqrt{2}}{2} \cos a + \frac{3\sqrt{2}}{2} \sin a + 1 = 0$$ 6. Combine like terms: $$\left(\frac{\sqrt{2}}{2} - \frac{3\sqrt{2}}{2}\right) \cos a + \left(-\frac{\sqrt{2}}{2} + \frac{3\sqrt{2}}{2}\right) \sin a + 1 = 0$$ $$-\sqrt{2} \cos a + \sqrt{2} \sin a + 1 = 0$$ 7. Divide the entire equation by $\sqrt{2}$ to simplify: $$-\cos a + \sin a + \frac{1}{\sqrt{2}} = 0$$ 8. Rearrange to isolate terms: $$\sin a - \cos a = -\frac{1}{\sqrt{2}}$$ 9. Use the identity $\sin a - \cos a = \sqrt{2} \sin\left(a - 45^\circ\right)$: $$\sqrt{2} \sin\left(a - 45^\circ\right) = -\frac{1}{\sqrt{2}}$$ 10. Divide both sides by $\sqrt{2}$: $$\sin\left(a - 45^\circ\right) = -\frac{1}{2}$$ 11. Solve for $a - 45^\circ$: $$a - 45^\circ = \arcsin\left(-\frac{1}{2}\right)$$ 12. The solutions for $\sin \theta = -\frac{1}{2}$ in degrees are: $$\theta = -30^\circ + 360^\circ k \quad \text{or} \quad \theta = 210^\circ + 360^\circ k, \quad k \in \mathbb{Z}$$ 13. Therefore: $$a - 45^\circ = -30^\circ + 360^\circ k \quad \Rightarrow \quad a = 15^\circ + 360^\circ k$$ $$a - 45^\circ = 210^\circ + 360^\circ k \quad \Rightarrow \quad a = 255^\circ + 360^\circ k$$ 14. Final answer: $$a = 15^\circ + 360^\circ k \quad \text{or} \quad a = 255^\circ + 360^\circ k, \quad k \in \mathbb{Z}$$