If two tangents inclined at an angle of `60^(@)` are drawn to a circle of radius 4 cm, then the length of each tangent is equal to :

Options

A. `2 sqrt(3)` cm
B. 8 cm
C. 4 cm
D. `4 sqrt(3)` cm

Answer

To solve the problem of finding the length of each tangent drawn to a circle of radius 4 cm, inclined at an angle of 60 degrees, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Geometry**:
- We have a circle with center O and radius 4 cm.
- Two tangents are drawn from a point P outside the circle, forming an angle of 60 degrees between them.

2. **Identify the Angles**:
- The angle between the two tangents (∠APB) is given as 60 degrees.
- Since the tangents are equal in length, we can bisect this angle. Thus, each angle (∠OAP and ∠OBP) will be half of 60 degrees, which is 30 degrees.

3. **Use the Right Triangle**:
- In triangle OAP, where OA is the radius (4 cm) and AP is the tangent we want to find, we know that ∠OAP = 30 degrees.
- The relationship between the sides of the triangle can be expressed using the tangent function:
\[ \tan(30^\circ) = \frac{OA}{AP} \]

4. **Substituting Known Values**:
- We know that \( \tan(30^\circ) = \frac{1}{\sqrt{3}} \).
- Substituting the known values into the equation gives:
\[ \frac{1}{\sqrt{3}} = \frac{4}{AP} \]

5. **Solving for AP**:
- Rearranging the equation to find AP:
\[ AP = 4 \cdot \sqrt{3} \]

6. **Final Answer**:
- Therefore, the length of each tangent is:
\[ AP = 4\sqrt{3} \text{ cm} \]

### Summary:
The length of each tangent drawn to the circle is \( 4\sqrt{3} \) cm.