1. Problem statement: Find the second angle $\alpha$ for the given trigonometric equations as in examples c) and d). 2. Important formulas and rules: - For sine: If $\sin \alpha = \sin \beta$, then the two solutions in $[0^\circ, 360^\circ)$ are $\alpha = \beta$ and $\alpha = 180^\circ - \beta$. - For cosine: If $\cos \alpha = \cos \beta$, then the two solutions in $[0^\circ, 360^\circ)$ are $\alpha = \beta$ and $\alpha = 360^\circ - \beta$. 3. Solve for 4c) $\sin \alpha = \sin 5^\circ$: - First solution: $\alpha = 5^\circ$ - Second solution: $\alpha = 180^\circ - 5^\circ = 175^\circ$ 4. Solve for 4d) $\cos \alpha = \cos 82^\circ$: - First solution: $\alpha = 82^\circ$ - Second solution: $\alpha = 360^\circ - 82^\circ = 278^\circ$ 5. Solve for 5a) $\sin \alpha = -\sin 23^\circ$: - Rewrite as $\sin \alpha = \sin (-23^\circ)$ since $-\sin 23^\circ = \sin (-23^\circ)$ - First solution: $\alpha = -23^\circ + 360^\circ = 337^\circ$ (to keep angle positive) - Second solution: $\alpha = 180^\circ - (-23^\circ) = 203^\circ$ 6. Solve for 5b) $\cos \alpha = -\cos 38^\circ$: - Rewrite as $\cos \alpha = \cos (180^\circ - 38^\circ) = \cos 142^\circ$ - First solution: $\alpha = 142^\circ$ - Second solution: $\alpha = 360^\circ - 142^\circ = 218^\circ$ 7. Solve for 5c) $\cos \alpha = -\cos 75^\circ$: - Rewrite as $\cos \alpha = \cos (180^\circ - 75^\circ) = \cos 105^\circ$ - First solution: $\alpha = 105^\circ$ - Second solution: $\alpha = 360^\circ - 105^\circ = 255^\circ$ 8. Solve for 5d) $\sin \alpha = -\sin 50^\circ$: - Rewrite as $\sin \alpha = \sin (-50^\circ)$ - First solution: $\alpha = -50^\circ + 360^\circ = 310^\circ$ - Second solution: $\alpha = 180^\circ - (-50^\circ) = 230^\circ$ Final answers: - 4c) $\alpha = 5^\circ$ or $175^\circ$ - 4d) $\alpha = 82^\circ$ or $278^\circ$ - 5a) $\alpha = 337^\circ$ or $203^\circ$ - 5b) $\alpha = 142^\circ$ or $218^\circ$ - 5c) $\alpha = 105^\circ$ or $255^\circ$ - 5d) $\alpha = 310^\circ$ or $230^\circ$