### Math Notes

Subjects

#### Trigonometry Solutions

##### Topics || Problems

A flagpole and a tower stand 36 meters apart on a horizontal plane. A person standing successively at their bases observes that the angle of elevation of the top of the tower is twice that of the pole, but at a point midway between their bases and angles of elevation are complementary. Find the height of the pole and tower.

$$\alpha + \beta = 90$$ and $$\delta + \beta = 90$$

$$\therefore \delta = \alpha$$

$$\tan \theta = \frac{P}{36}$$ (1)

$$\tan 2 \theta = \frac{T}{36}$$ (2)

$$\tan \alpha = \frac{T}{18}$$ (3)

$$\tan \alpha = \frac{18}{P}$$ (4)

Combine (3) and (4)

$$T = \frac{18^2}{P}$$ (3.4)

Simplify (2)

$$\tan 2\theta = \frac{2 \tan \theta}{1-tan^2 \theta}$$

$$\frac{T}{36} = \frac{2 \tan \theta}{1-tan^2 \theta}$$

Combine simplified (2) and (1)

$$\frac{T}{36}= \frac{2 \frac{P}{36}}{1-(\frac{P}{36})^2}$$

$$\frac{T}{36}= \frac{72P}{36^2 - P^2}$$

$$P = \frac{2592P}{36^2-P}$$ BUT $$T = \frac{18^2}{P}$$

Thus

$$\frac{18^2}{P} = \frac{2592P}{36^2-P}$$

$$P^2 = \frac{36^2-P^2}{8}$$

$$9P^2 = 162 (8)$$

$$P = 12 m$$

$$T = \frac{18^2}{12}$$

$$T = 27 m$$

Therefore: The height of the pole is 12 meters and the height of the tower is 27 meters.