If `angleA=60^(@), b=2` and c=4, then the value of a is

Options

A. `2sqrt(2)`
B. `3sqrt(2)`
C. `2sqrt(3)`
D. `sqrt(6)`

Answer

To find the value of side \( a \) in the triangle given \( \angle A = 60^\circ \), \( b = 2 \), and \( c = 4 \), 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 formula.

Given:
- \( A = 60^\circ \)
- \( b = 2 \)
- \( c = 4 \)

We substitute these values into the formula:

\[ \cos 60^\circ = \frac{2^2 + 4^2 - a^2}{2 \cdot 2 \cdot 4} \]

### Step 2: Calculate \( \cos 60^\circ \).

We know that:

\[ \cos 60^\circ = \frac{1}{2} \]

### Step 3: Substitute \( \cos 60^\circ \) into the equation.

Now, we can rewrite the equation:

\[ \frac{1}{2} = \frac{2^2 + 4^2 - a^2}{2 \cdot 2 \cdot 4} \]

### Step 4: Simplify the equation.

Calculate \( 2^2 \) and \( 4^2 \):

\[ 2^2 = 4 \quad \text{and} \quad 4^2 = 16 \]

So, we have:

\[ \frac{1}{2} = \frac{4 + 16 - a^2}{16} \]

This simplifies to:

\[ \frac{1}{2} = \frac{20 - a^2}{16} \]

### Step 5: Cross-multiply to eliminate the fraction.

Cross-multiplying gives:

\[ 1 \cdot 16 = 2(20 - a^2) \]

This simplifies to:

\[ 16 = 40 - 2a^2 \]

### Step 6: Rearrange the equation to solve for \( a^2 \).

Rearranging gives:

\[ 2a^2 = 40 - 16 \]

\[ 2a^2 = 24 \]

### Step 7: Divide by 2 to isolate \( a^2 \).

\[ a^2 = \frac{24}{2} \]

\[ a^2 = 12 \]

### Step 8: Take the square root to find \( a \).

\[ a = \sqrt{12} \]

This can be simplified to:

\[ a = 2\sqrt{3} \]

### Step 9: Conclude with the positive value.

Since \( a \) represents a side of a triangle, we take the positive value:

\[ a = 2\sqrt{3} \]

### Final Answer:
The value of \( a \) is \( 2\sqrt{3} \).

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