1. **Problem Statement:** Find the minimum value of $$2 \cos \theta + \frac{1}{\sin \theta} + \sqrt{2} \tan \theta$$ where $$\theta$$ is an acute angle. 2. **Understanding the problem:** We want to minimize the sum of three terms involving trigonometric functions of $$\theta$$. 3. **Why use AM-GM inequality?** - The Arithmetic Mean - Geometric Mean (AM-GM) inequality states that for any non-negative real numbers $$a, b, c$$: $$\frac{a+b+c}{3} \geq \sqrt[3]{abc}$$ - Equality holds if and only if $$a = b = c$$. - Since the terms $$2 \cos \theta$$, $$\frac{1}{\sin \theta}$$, and $$\sqrt{2} \tan \theta$$ are positive for acute $$\theta$$, AM-GM is applicable. - Using AM-GM helps find a lower bound for the sum and identify when the minimum occurs. 4. **Applying AM-GM:** $$\frac{2 \cos \theta + \frac{1}{\sin \theta} + \sqrt{2} \tan \theta}{3} \geq \sqrt[3]{2 \cos \theta \cdot \frac{1}{\sin \theta} \cdot \sqrt{2} \tan \theta}$$ 5. **Simplify the right side:** - Recall $$\tan \theta = \frac{\sin \theta}{\cos \theta}$$ - Substitute: $$2 \cos \theta \cdot \frac{1}{\sin \theta} \cdot \sqrt{2} \cdot \frac{\sin \theta}{\cos \theta} = 2 \cos \theta \cdot \frac{1}{\sin \theta} \cdot \sqrt{2} \cdot \frac{\sin \theta}{\cos \theta}$$ - Cancel terms: $$= 2 \cdot \sqrt{2} = 2\sqrt{2}$$ 6. **Therefore:** $$\frac{2 \cos \theta + \frac{1}{\sin \theta} + \sqrt{2} \tan \theta}{3} \geq \sqrt[3]{2\sqrt{2}}$$ 7. **Calculate the cube root:** $$\sqrt[3]{2\sqrt{2}} = \sqrt[3]{2 \cdot 2^{1/2}} = \sqrt[3]{2^{3/2}} = 2^{\frac{3}{2} \cdot \frac{1}{3}} = 2^{1/2} = \sqrt{2}$$ 8. **Multiply both sides by 3:** $$2 \cos \theta + \frac{1}{\sin \theta} + \sqrt{2} \tan \theta \geq 3 \sqrt{2}$$ 9. **Conclusion:** - The minimum value of the expression is $$3 \sqrt{2}$$. - This minimum occurs when all three terms are equal: $$2 \cos \theta = \frac{1}{\sin \theta} = \sqrt{2} \tan \theta$$. **Summary:** - We used the AM-GM inequality because it provides a way to find the minimum sum of positive terms. - The key logic is that the minimum sum occurs when all terms are equal, which is the equality condition of AM-GM.