1. **State the problem:** Find the real solutions for the equations: (a) $3 \sin \theta - 4 \cos \theta = 2$, (b) $4 \sin \theta + 3 \cos \theta = 6$, (c) $12 \sin \theta - 5 \cos \theta = -6$. 2. **Formula and method:** Each equation is of the form $a \sin \theta + b \cos \theta = c$. We use the identity: $$a \sin \theta + b \cos \theta = R \sin(\theta + \alpha)$$ where $$R = \sqrt{a^2 + b^2}$$ and $$\alpha = \arctan\left(\frac{b}{a}\right)$$ This transforms the equation into: $$R \sin(\theta + \alpha) = c$$ which can be solved for $\theta$ if $|c| \leq R$. 3. **Solve (a):** - $a=3$, $b=-4$, $c=2$ - Calculate $R = \sqrt{3^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5$ - Calculate $\alpha = \arctan\left(\frac{-4}{3}\right)$ 4. **Rewrite equation (a):** $$5 \sin(\theta + \alpha) = 2$$ $$\sin(\theta + \alpha) = \frac{2}{5}$$ 5. **Find solutions for (a):** $$\theta + \alpha = \sin^{-1}\left(\frac{2}{5}\right) \quad \text{or} \quad \pi - \sin^{-1}\left(\frac{2}{5}\right)$$ 6. **Calculate $\alpha$ and $\sin^{-1}(2/5)$ numerically:** - $\alpha = \arctan(-\frac{4}{3}) \approx -0.9273$ - $\sin^{-1}(\frac{2}{5}) \approx 0.4115$ 7. **Final solutions for (a):** $$\theta = 0.4115 - (-0.9273) = 1.3388$$ $$\theta = \pi - 0.4115 - (-0.9273) = 3.6574$$ 8. **Solve (b):** - $a=4$, $b=3$, $c=6$ - Calculate $R = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5$ - Calculate $\alpha = \arctan\left(\frac{3}{4}\right) \approx 0.6435$ 9. **Rewrite equation (b):** $$5 \sin(\theta + 0.6435) = 6$$ $$\sin(\theta + 0.6435) = \frac{6}{5}$$ Since $\frac{6}{5} = 1.2 > 1$, no real solutions exist for (b). 10. **Solve (c):** - $a=12$, $b=-5$, $c=-6$ - Calculate $R = \sqrt{12^2 + (-5)^2} = \sqrt{144 + 25} = \sqrt{169} = 13$ - Calculate $\alpha = \arctan\left(\frac{-5}{12}\right) \approx -0.3948$ 11. **Rewrite equation (c):** $$13 \sin(\theta - 0.3948) = -6$$ $$\sin(\theta - 0.3948) = \frac{-6}{13} = -0.4615$$ 12. **Find solutions for (c):** $$\theta - 0.3948 = \sin^{-1}(-0.4615) \quad \text{or} \quad \pi - \sin^{-1}(-0.4615)$$ 13. **Calculate $\sin^{-1}(-0.4615)$ numerically:** $$\sin^{-1}(-0.4615) \approx -0.4791$$ 14. **Final solutions for (c):** $$\theta = -0.4791 + 0.3948 = -0.0843$$ $$\theta = \pi - (-0.4791) + 0.3948 = 3.6155$$ **Summary:** - (a) $\theta \approx 1.3388, 3.6574$ - (b) No real solutions - (c) $\theta \approx -0.0843, 3.6155$