If `5costheta+3cos(theta+(pi)/(3))+"sin"^(2)(pi)/(3)-2=A` then value of A will be between ?

Options

A. `[-7,7]`
B. `[5(3)/(4),-8(1)/(4)]`
C. `[-5(3)/(4),+8(1)/(4)]`
D. `[-8(1)/(4),+5(3)/(4)]`

Answer

To solve the equation \( 5 \cos \theta + 3 \cos \left( \theta + \frac{\pi}{3} \right) + \sin^2 \left( \frac{\pi}{3} \right) - 2 = A \), we will follow these steps:

### Step 1: Calculate \( \sin^2 \left( \frac{\pi}{3} \right) \)
We know that:
\[ \sin \left( \frac{\pi}{3} \right) = \frac{\sqrt{3}}{2} \]
Thus,
\[ \sin^2 \left( \frac{\pi}{3} \right) = \left( \frac{\sqrt{3}}{2} \right)^2 = \frac{3}{4} \]

### Step 2: Rewrite \( 3 \cos \left( \theta + \frac{\pi}{3} \right) \)
Using the cosine addition formula:
\[ \cos \left( a + b \right) = \cos a \cos b - \sin a \sin b \]
we can write:
\[ 3 \cos \left( \theta + \frac{\pi}{3} \right) = 3 \left( \cos \theta \cos \frac{\pi}{3} - \sin \theta \sin \frac{\pi}{3} \right) \]
Substituting \( \cos \frac{\pi}{3} = \frac{1}{2} \) and \( \sin \frac{\pi}{3} = \frac{\sqrt{3}}{2} \):
\[ = 3 \left( \cos \theta \cdot \frac{1}{2} - \sin \theta \cdot \frac{\sqrt{3}}{2} \right) = \frac{3}{2} \cos \theta - \frac{3\sqrt{3}}{2} \sin \theta \]

### Step 3: Substitute back into the equation
Now substituting back into the equation for \( A \):
\[ A = 5 \cos \theta + \left( \frac{3}{2} \cos \theta - \frac{3\sqrt{3}}{2} \sin \theta \right) + \frac{3}{4} - 2 \]
Combining like terms:
\[ A = \left( 5 + \frac{3}{2} \right) \cos \theta - \frac{3\sqrt{3}}{2} \sin \theta + \frac{3}{4} - 2 \]
Calculating \( \frac{3}{4} - 2 \):
\[ = \frac{3}{4} - \frac{8}{4} = -\frac{5}{4} \]
Thus,
\[ A = \frac{13}{2} \cos \theta - \frac{3\sqrt{3}}{2} \sin \theta - \frac{5}{4} \]

### Step 4: Find the range of \( A \)
To find the maximum and minimum values of \( A \), we can express it in the form:
\[ A = R \cos(\theta + \phi) - \frac{5}{4} \]
where \( R = \sqrt{\left( \frac{13}{2} \right)^2 + \left( \frac{3\sqrt{3}}{2} \right)^2} \).

Calculating \( R \):
\[ R = \sqrt{\frac{169}{4} + \frac{27}{4}} = \sqrt{\frac{196}{4}} = \sqrt{49} = 7 \]
Thus,
\[ A = 7 \cos(\theta + \phi) - \frac{5}{4} \]

### Step 5: Determine the range
Since \( \cos(\theta + \phi) \) varies between -1 and 1:
- Minimum value of \( A \):
\[ A_{\text{min}} = 7(-1) - \frac{5}{4} = -7 - \frac{5}{4} = -\frac{28}{4} - \frac{5}{4} = -\frac{33}{4} \]
- Maximum value of \( A \):
\[ A_{\text{max}} = 7(1) - \frac{5}{4} = 7 - \frac{5}{4} = \frac{28}{4} - \frac{5}{4} = \frac{23}{4} \]

### Conclusion
Thus, the value of \( A \) will be between:
\[ -\frac{33}{4} \text{ and } \frac{23}{4} \]