The maximum value of $5~cos~ heta+3~cos( heta+\frac{\pi}{3})+3$ is:

Q: The maximum value of $5~cos~\theta+3~cos(\theta+\frac{\pi}{3})+3$ is:

The maximum value of $5~cos~\theta+3~cos(\theta+\frac{\pi}{3})+3$ is:

Question

The maximum value of $5~cos~	heta+3~cos(	heta+\frac{\pi}{3})+3$ is:

Accepted Answer

To determine the maximum value of the expression $5 \cos \theta + 3 \cos \left(\theta + \frac{\pi}{3}\right) + 3$, we need to analyze the trigonometric components and their possible maximum values.

1. **Expression Analysis:**
   - The expression is $5 \cos \theta + 3 \cos \left(\theta + \frac{\pi}{3}\right) + 3$.
   - We know that the maximum value of $\cos \theta$ is 1.

2. **Simplifying the Expression:**
   - Let's rewrite $\cos \left(\theta + \frac{\pi}{3}\right)$ using the angle addition formula:
     $$
     \cos \left(\theta + \frac{\pi}{3}\right) = \cos \theta \cos \frac{\pi}{3} - \sin \theta \sin \frac{\pi}{3}
     $$
   - Substituting the values $\cos \frac{\pi}{3} = \frac{1}{2}$ and $\sin \frac{\pi}{3} = \frac{\sqrt{3}}{2}$, we get:
     $$
     \cos \left(\theta + \frac{\pi}{3}\right) = \frac{1}{2} \cos \theta - \frac{\sqrt{3}}{2} \sin \theta
     $$

3. **Substitute and Simplify:**
   - Substitute back into the original expression:
     $$
     5 \cos \theta + 3 \left(\frac{1}{2} \cos \theta - \frac{\sqrt{3}}{2} \sin \theta\right) + 3
     $$
   - Simplify:
     $$
     5 \cos \theta + \frac{3}{2} \cos \theta - \frac{3\sqrt{3}}{2} \sin \theta + 3
     $$
   - Combine like terms:
     $$
     \left(5 + \frac{3}{2}\right) \cos \theta - \frac{3\sqrt{3}}{2} \sin \theta + 3 = \frac{13}{2} \cos \theta - \frac{3\sqrt{3}}{2} \sin \theta + 3
     $$

4. **Maximizing the Expression:**
   - The expression $\frac{13}{2} \cos \theta - \frac{3\sqrt{3}}{2} \sin \theta$ can be rewritten in the form $R \cos(\theta + \alpha)$, where $R = \sqrt{\left(\frac{13}{2}\right)^2 + \left(\frac{3\sqrt{3}}{2}\right)^2}$.
   - Calculate $R$:
     $$
     R = \sqrt{\left(\frac{13}{2}\right)^2 + \left(\frac{3\sqrt{3}}{2}\right)^2} = \sqrt{\frac{169}{4} + \frac{27}{4}} = \sqrt{\frac{196}{4}} = \sqrt{49} = 7
     $$
   - The maximum value of $R \cos(\theta + \alpha)$ is $R$, which is 7.

5. **Final Calculation:**
   - Therefore, the maximum value of the entire expression is:
     $$
     R + 3 = 7 + 3 = 10
     $$

Thus, the maximum value of the expression is 10, which corresponds to Option 2.

**Explanation for Other Options:**
- **Option 1 (8):** This value is less than the calculated maximum of 10.
- **Option 3 (11):** This value exceeds the calculated maximum of 10.
- **Option 4 (7):** This value is the maximum of the trigonometric part without adding the constant 3, hence less than the total maximum of 10.

Therefore, Option 2 is the correct answer, as it matches the calculated maximum value of the expression.