Given that `y=(10)/(sinx+sqrt(3)cosx)`.Minimum value of y is

Options

A. zero
B. 2
C. 5
D. `10//(1+sqrt()3)`

Answer

To find the minimum value of \( y = \frac{10}{\sin x + \sqrt{3} \cos x} \), we can follow these steps:

### Step 1: Analyze the Denominator
The expression \( \sin x + \sqrt{3} \cos x \) is the denominator of \( y \). To minimize \( y \), we need to maximize the denominator.

### Step 2: Rewrite the Denominator
We can rewrite \( \sin x + \sqrt{3} \cos x \) in a more manageable form. We can express it as:
\[ R \sin(x + \phi) \]
where \( R = \sqrt{1^2 + (\sqrt{3})^2} = \sqrt{4} = 2 \) and \( \phi \) is the angle such that \( \tan \phi = \frac{\sqrt{3}}{1} \), which gives \( \phi = 60^\circ \) or \( \frac{\pi}{3} \).

Thus, we have:
\[ \sin x + \sqrt{3} \cos x = 2 \sin\left(x + \frac{\pi}{3}\right) \]

### Step 3: Find Maximum Value of the Denominator
The maximum value of \( \sin\left(x + \frac{\pi}{3}\right) \) is \( 1 \). Therefore, the maximum value of the denominator \( \sin x + \sqrt{3} \cos x \) is:
\[ 2 \cdot 1 = 2 \]

### Step 4: Substitute Back to Find Minimum Value of \( y \)
Now substituting this maximum value back into the expression for \( y \):
\[ y = \frac{10}{\sin x + \sqrt{3} \cos x} \implies y_{\text{min}} = \frac{10}{2} = 5 \]

### Conclusion
The minimum value of \( y \) is \( 5 \).

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