A person standing at the foot of a tower walks a distance of 3 meters from the tower and observes that the angle of elevation of the top of the tower is `30^(@)`. He then walks a distance 4 meters perpendicular to the previous direction and observes the angle of elevation to be `beta`. Then, `cos 2beta` is equal to

Options

A. `(sqrt3)/(2)`
B. `(1)/(sqrt3)`
C. `(2)/(sqrt3)`
D. `(11)/(14)`

Answer

To solve the problem step by step, we will follow the given information and apply trigonometric principles.

### Step 1: Understand the setup
A person is standing at the foot of a tower and walks 3 meters away from it. The angle of elevation to the top of the tower at this point is \(30^\circ\).

### Step 2: Identify the triangle
Let:
- \(O\) be the foot of the tower,
- \(A\) be the point where the person is standing after walking 3 meters away from the tower,
- \(B\) be the top of the tower.

In triangle \(OAB\):
- \(OA = 3\) meters (horizontal distance from the tower),
- \(AB = h\) meters (height of the tower),
- Angle \(AOB = 30^\circ\).

### Step 3: Use the tangent function
Using the tangent of the angle of elevation:
\[ \tan(30^\circ) = \frac{h}{OA} \]
Substituting the known values:
\[ \tan(30^\circ) = \frac{h}{3} \]
We know that \(\tan(30^\circ) = \frac{1}{\sqrt{3}}\):
\[ \frac{1}{\sqrt{3}} = \frac{h}{3} \]

### Step 4: Solve for \(h\)
Cross-multiplying gives:
\[ h = \frac{3}{\sqrt{3}} = \sqrt{3} \text{ meters} \]

### Step 5: Determine the new position
The person then walks 4 meters perpendicular to the previous direction. Let \(P\) be the new position of the person. The distance \(OP\) can be calculated using the Pythagorean theorem:
\[ OP = \sqrt{OA^2 + AP^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \text{ meters} \]

### Step 6: Use the tangent function again
In triangle \(OPB\), we can again use the tangent function:
\[ \tan(\beta) = \frac{h}{OP} = \frac{\sqrt{3}}{5} \]

### Step 7: Calculate \(\tan^2(\beta)\)
Now, squaring both sides:
\[ \tan^2(\beta) = \left(\frac{\sqrt{3}}{5}\right)^2 = \frac{3}{25} \]

### Step 8: Use the double angle formula for cosine
We need to find \(\cos(2\beta)\). The double angle formula is:
\[ \cos(2\beta) = \frac{1 - \tan^2(\beta)}{1 + \tan^2(\beta)} \]
Substituting the value of \(\tan^2(\beta)\):
\[ \cos(2\beta) = \frac{1 - \frac{3}{25}}{1 + \frac{3}{25}} = \frac{\frac{25 - 3}{25}}{\frac{25 + 3}{25}} = \frac{22/25}{28/25} = \frac{22}{28} = \frac{11}{14} \]

### Final Answer
Thus, the value of \(\cos(2\beta)\) is \(\frac{11}{14}\).

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