If `alpha + beta = 90^@ and alpha: beta = 2: 1`, then the value of `sin alpha : sin beta` is

Options

A. `sqrt3 : 1`
B. `2 : 1`
C. `1 : 1`
D. `sqrt2:1`

Answer

To solve the problem, we need to find the value of \( \sin \alpha : \sin \beta \) given that \( \alpha + \beta = 90^\circ \) and the ratio \( \alpha : \beta = 2 : 1 \).

### Step-by-step Solution:

1. **Set up the equations based on the given ratio**:
Since \( \alpha : \beta = 2 : 1 \), we can express \( \alpha \) and \( \beta \) in terms of a variable \( x \):
\[ \alpha = 2x \quad \text{and} \quad \beta = x \]

2. **Use the sum of angles**:
We know that \( \alpha + \beta = 90^\circ \). Substituting the expressions for \( \alpha \) and \( \beta \):
\[ 2x + x = 90^\circ \]
Simplifying this gives:
\[ 3x = 90^\circ \]

3. **Solve for \( x \)**:
Dividing both sides by 3:
\[ x = 30^\circ \]

4. **Find the values of \( \alpha \) and \( \beta \)**:
Now substituting back to find \( \alpha \) and \( \beta \):
\[ \beta = x = 30^\circ \]
\[ \alpha = 2x = 2 \times 30^\circ = 60^\circ \]

5. **Calculate \( \sin \alpha \) and \( \sin \beta \)**:
Now we find the sine values:
\[ \sin \alpha = \sin 60^\circ = \frac{\sqrt{3}}{2} \]
\[ \sin \beta = \sin 30^\circ = \frac{1}{2} \]

6. **Set up the ratio \( \sin \alpha : \sin \beta \)**:
Now we can express the ratio:
\[ \sin \alpha : \sin \beta = \frac{\sqrt{3}}{2} : \frac{1}{2} \]

7. **Simplify the ratio**:
Dividing both sides by \( \frac{1}{2} \):
\[ \sin \alpha : \sin \beta = \sqrt{3} : 1 \]

### Final Answer:
The value of \( \sin \alpha : \sin \beta \) is \( \sqrt{3} : 1 \).

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