In a triangle ABC, if a=4, b=8 and `angleC=60^(@)`, then :
(where symbols used have usual meanings)

Options

A. `c=6`
B. `c=4sqrt(3)`
C. `angleA=30^(@)`
D. `angleB=90^(@)`

Answer

To solve the triangle ABC with given values \( a = 4 \), \( b = 8 \), and \( \angle C = 60^\circ \), we will follow these steps:

### Step 1: Use the Cosine Rule to find side \( c \)

The Cosine Rule states:
\[ c^2 = a^2 + b^2 - 2ab \cos(C) \]

Substituting the known values:
\[ c^2 = 4^2 + 8^2 - 2 \cdot 4 \cdot 8 \cdot \cos(60^\circ) \]
\[ c^2 = 16 + 64 - 2 \cdot 4 \cdot 8 \cdot \frac{1}{2} \]
\[ c^2 = 16 + 64 - 32 \]
\[ c^2 = 48 \]
\[ c = \sqrt{48} = 4\sqrt{3} \]

### Step 2: Use the Sine Rule to find angle \( A \)

The Sine Rule states:
\[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \]

Using the first part of the Sine Rule:
\[ \frac{4}{\sin A} = \frac{4\sqrt{3}}{\sin(60^\circ)} \]

Since \( \sin(60^\circ) = \frac{\sqrt{3}}{2} \), we can substitute:
\[ \frac{4}{\sin A} = \frac{4\sqrt{3}}{\frac{\sqrt{3}}{2}} \]
\[ \frac{4}{\sin A} = 8 \]
Cross-multiplying gives:
\[ 4 = 8 \sin A \]
\[ \sin A = \frac{4}{8} = \frac{1}{2} \]
Thus:
\[ A = 30^\circ \]

### Step 3: Find angle \( B \)

Since the sum of angles in a triangle is \( 180^\circ \):
\[ A + B + C = 180^\circ \]
Substituting the known values:
\[ 30^\circ + B + 60^\circ = 180^\circ \]
\[ B = 180^\circ - 90^\circ = 90^\circ \]

### Final Results

- The length of side \( c \) is \( 4\sqrt{3} \).
- Angle \( A \) is \( 30^\circ \).
- Angle \( B \) is \( 90^\circ \).
- Angle \( C \) is \( 60^\circ \).