The angles of elevation of the top of a tower standing on a horizontal plane from two points on a line passing through the foot of the tower at distances a and b respectively are complementry angles. If the line joning the two points subtend an angle `theta` at the top of the tower then `h=`____and `sintheta=`____.

Answer

To solve the problem, we need to derive the height of the tower (h) and the sine of the angle (sin θ) based on the given conditions. Let's break it down step by step.

### Step 1: Understand the Geometry
We have a tower PQ, and two points A and B on the ground at distances a and b from the foot of the tower (point O). The angles of elevation from points A and B to the top of the tower are complementary angles, meaning if one angle is θ, the other angle is (90° - θ).

### Step 2: Set Up the Right Triangles
From point A:
- Let the angle of elevation to the top of the tower be θ.
- The height of the tower is h.
- The distance from point A to the foot of the tower is a.

Using the tangent function:
\[ \tan(\theta) = \frac{h}{a} \quad \text{(1)} \]

From point B:
- The angle of elevation to the top of the tower is (90° - θ).
- The distance from point B to the foot of the tower is b.

Using the tangent function:
\[ \tan(90° - \theta) = \cot(\theta) = \frac{h}{b} \quad \text{(2)} \]

### Step 3: Relate the Two Equations
From equation (1):
\[ h = a \tan(\theta) \quad \text{(3)} \]

From equation (2):
\[ h = b \cot(\theta) \quad \text{(4)} \]

### Step 4: Equate the Two Expressions for h
Setting equations (3) and (4) equal to each other:
\[ a \tan(\theta) = b \cot(\theta) \]

### Step 5: Substitute Cotangent
Recall that \(\cot(\theta) = \frac{1}{\tan(\theta)}\):
\[ a \tan(\theta) = \frac{b}{\tan(\theta)} \]

### Step 6: Cross Multiply
Cross multiplying gives:
\[ a \tan^2(\theta) = b \]

### Step 7: Solve for tan(θ)
Rearranging gives:
\[ \tan^2(\theta) = \frac{b}{a} \]
Taking the square root:
\[ \tan(\theta) = \sqrt{\frac{b}{a}} \quad \text{(5)} \]

### Step 8: Substitute Back to Find h
Substituting (5) back into equation (3):
\[ h = a \tan(\theta) = a \sqrt{\frac{b}{a}} = \sqrt{ab} \]

### Step 9: Find sin(θ)
Using the Pythagorean identity:
\[ \sin^2(\theta) + \cos^2(\theta) = 1 \]
We know:
\[ \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} = \sqrt{\frac{b}{a}} \]
Thus:
\[ \sin(\theta) = \tan(\theta) \cos(\theta) = \sqrt{\frac{b}{a}} \cos(\theta) \]

Using the identity:
\[ \cos^2(\theta) = \frac{1}{1 + \tan^2(\theta)} = \frac{a}{a + b} \]
So:
\[ \sin(\theta) = \sqrt{\frac{b}{a}} \cdot \sqrt{\frac{a}{a + b}} = \frac{\sqrt{ab}}{\sqrt{a + b}} \]

### Final Answers
Thus, we have:
- Height of the tower \( h = \sqrt{ab} \)
- Sine of the angle \( \sin(\theta) = \frac{\sqrt{ab}}{\sqrt{a + b}} \)