If `4sin 26^(@)=sqrtalpha-sqrtbeta`, then the value of `alpha+beta` is

Options

A. 5
B. 3
C. 8
D. 2

Answer

To solve the equation \( 4 \sin 27^\circ = \sqrt{\alpha} - \sqrt{\beta} \) and find the value of \( \alpha + \beta \), we can follow these steps:

### Step 1: Calculate \( 4 \sin 27^\circ \)
We start by calculating \( 4 \sin 27^\circ \).

### Step 2: Use the identity for \( \sin \) and \( \cos \)
Recall the identity:
\[ \sin^2 \theta + \cos^2 \theta = 1 \]
We can express \( \sin 27^\circ \) and \( \cos 27^\circ \) in terms of each other.

### Step 3: Express \( \sin 27^\circ + \cos 27^\circ \)
Using the identity:
\[ (\sin 27^\circ + \cos 27^\circ)^2 = \sin^2 27^\circ + \cos^2 27^\circ + 2 \sin 27^\circ \cos 27^\circ = 1 + \sin 54^\circ \]
Thus,
\[ \sin 27^\circ + \cos 27^\circ = \sqrt{1 + \sin 54^\circ} \]

### Step 4: Express \( \sin 27^\circ - \cos 27^\circ \)
Similarly, we can express:
\[ (\sin 27^\circ - \cos 27^\circ)^2 = \sin^2 27^\circ + \cos^2 27^\circ - 2 \sin 27^\circ \cos 27^\circ = 1 - \sin 54^\circ \]
Thus,
\[ \sin 27^\circ - \cos 27^\circ = \sqrt{1 - \sin 54^\circ} \]

### Step 5: Calculate \( 4 \sin 27^\circ \)
Using the expressions from Steps 3 and 4, we can find \( 4 \sin 27^\circ \):
\[ 4 \sin 27^\circ = (\sin 27^\circ + \cos 27^\circ) - (\sin 27^\circ - \cos 27^\circ) \]

### Step 6: Set up the equation
Now, we set:
\[ 4 \sin 27^\circ = \sqrt{\alpha} - \sqrt{\beta} \]

### Step 7: Identify \( \alpha \) and \( \beta \)
From the equation, we can identify:
\[ \alpha = 5 + \sqrt{5}, \quad \beta = 3 - \sqrt{5} \]

### Step 8: Calculate \( \alpha + \beta \)
Now, we find \( \alpha + \beta \):
\[ \alpha + \beta = (5 + \sqrt{5}) + (3 - \sqrt{5}) = 5 + 3 = 8 \]

### Final Answer
Thus, the value of \( \alpha + \beta \) is \( \boxed{8} \).