What is the value of `beta - alpha`, if sin `alpha = (sqrt3)/(2) and cos beta = 0`?

Options

A. `0^(@)`
B. `30^(@)`
C. `45^(@)`
D. `60^(@)`

Answer

To find the value of \( \beta - \alpha \) given that \( \sin \alpha = \frac{\sqrt{3}}{2} \) and \( \cos \beta = 0 \), we can follow these steps:

### Step 1: Determine the value of \( \alpha \)
We know that \( \sin \alpha = \frac{\sqrt{3}}{2} \). The sine function equals \( \frac{\sqrt{3}}{2} \) at specific angles.

The angle where \( \sin \alpha = \frac{\sqrt{3}}{2} \) is:
\[ \alpha = 60^\circ \quad \text{(since } \sin 60^\circ = \frac{\sqrt{3}}{2}\text{)} \]

### Step 2: Determine the value of \( \beta \)
Next, we know that \( \cos \beta = 0 \). The cosine function equals 0 at specific angles as well.

The angle where \( \cos \beta = 0 \) is:
\[ \beta = 90^\circ \quad \text{(since } \cos 90^\circ = 0\text{)} \]

### Step 3: Calculate \( \beta - \alpha \)
Now that we have both \( \alpha \) and \( \beta \), we can find \( \beta - \alpha \):
\[ \beta - \alpha = 90^\circ - 60^\circ = 30^\circ \]

### Final Answer
Thus, the value of \( \beta - \alpha \) is:
\[ \boxed{30^\circ} \]

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