In a triangle ABC, b = `sqrt(3)` cm, c = 1 cm, `angle A = 30^(@)`, what is the value of a ?

Options

A. `sqrt(2)` cm
B. 2 cm
C. 1 cm
D. `(1)/(2)` cm

Explanation

To find the value of side \( a \) in triangle \( ABC \) where \( b = \sqrt{3} \) cm, \( c = 1 \) cm, and \( \angle A = 30^\circ \), we can use the cosine rule. The cosine rule states:

\[ \cos A = \frac{b^2 + c^2 - a^2}{2bc} \]

### Step 1: Substitute the known values into the cosine rule.

Given:
- \( b = \sqrt{3} \)
- \( c = 1 \)
- \( \angle A = 30^\circ \)

We know that \( \cos 30^\circ = \frac{\sqrt{3}}{2} \).

Now substituting the values into the cosine rule:

\[ \frac{\sqrt{3}}{2} = \frac{(\sqrt{3})^2 + (1)^2 - a^2}{2 \cdot \sqrt{3} \cdot 1} \]

### Step 2: Simplify the equation.

Calculating \( b^2 \) and \( c^2 \):

\[ (\sqrt{3})^2 = 3 \quad \text{and} \quad (1)^2 = 1 \]

So, the equation becomes:

\[ \frac{\sqrt{3}}{2} = \frac{3 + 1 - a^2}{2\sqrt{3}} \]

This simplifies to:

\[ \frac{\sqrt{3}}{2} = \frac{4 - a^2}{2\sqrt{3}} \]

### Step 3: Cross-multiply to eliminate the fraction.

Cross-multiplying gives:

\[ \sqrt{3} \cdot 2\sqrt{3} = 2(4 - a^2) \]

This simplifies to:

\[ 6 = 8 - 2a^2 \]

### Step 4: Rearrange the equation to solve for \( a^2 \).

Rearranging gives:

\[ 2a^2 = 8 - 6 \]

So,

\[ 2a^2 = 2 \]

Dividing both sides by 2:

\[ a^2 = 1 \]

### Step 5: Take the square root to find \( a \).

Taking the square root of both sides:

\[ a = 1 \text{ cm} \]

### Conclusion:

The value of \( a \) is \( 1 \) cm.

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