1. From a point 150 meters away from the base of an incomplete pillar, the angle of elevation to its top is 45°. How much additional height must be added to the pillar so that, from the same point, the angle of elevation to the new top becomes 60°?

2. A person is positioned along a horizontal straight line on the same plane as a chimney. From a certain point, the angle of elevation to the top of the chimney is 30°. After moving 50 meters closer along the same straight path, the angle of elevation becomes 60°. Calculate and write the height of the chimney.

3. The distance between two pillars is 150 meters. One is three times taller than the other. From the midpoint of the line segment connecting the bases of the two pillars, the angles of elevation to the tops of the pillars are complementary. Find the height of the shorter pillar.

4. If the angle of elevation to the top of a pillar is 45° from a point 20 meters away from its base, then the height of the pillar is —

(a) 10 meters (b) 20 meters (c) 40 meters (d) 50 meters

5. If the angle of elevation of the top of a chimney from a point on level ground at its base is 60°, and from another point on the same straight line 24 meters further away from the base, the angle of elevation is 30°, then calculate and write the height of the chimney. [Take the approximate value of \(\sqrt{3} = 1.732\) and find the result correct to three decimal places.]

6. From a point on the roof of a five-storey building, the angle of elevation to the top of a monument is 60°, and the angle of depression to its base is 30°. If the height of the building is 16 meters, find the height of the monument and the distance from the building to the monument.

7. Two points on the ground lie along the same straight line with the base of a vertical pillar. From these two points, the angles of elevation to the top of the pillar are complementary. If the distances from the two points to the base of the pillar are 9 meters and 16 meters respectively, and both points are on the same side of the pillar, find the height of the pillar.

8. From a point on the roof of a five-story building, the elevation angle to the top of a monument is 60°, and the depression angle to its base is 30°. If the building's height is 16 meters, determine the height of the monument and its distance from the building.

9. The distance between two pillars is 150 meters. One pillar's height is three times the other. The angles of elevation to their tops from the midpoint of the line joining their bases are complementary. Find the height of the smaller pillar.

10. The distance between two pillars is 150 meters. One is three times as tall as the other. From the midpoint of the line connecting the bases of the two pillars, the angles of elevation to their tops are complementary. What is the height of the shorter pillar?

11. From a point on the roof of a five-storied building, the angle of elevation to the top of a monument and the angle of depression to its base are 60° and 30° respectively. If the height of the building is 16 meters, find the height of the monument and the horizontal distance between the building and the monument.

12. If the angle of elevation of the top of a coconut tree from a point on the horizontal ground 20 meters away from its base is 60°, then find the height of the tree.

13. From a point on the roof of a five-story building, the angle of elevation to the top of a monument is 60°, and the angle of depression to its base is 30°. If the height of the building is 16 meters, then calculate and write the height of the monument and the distance of the building from the monument.

14. From two points located on the same side and along the same horizontal line passing through the base of a vertical pillar, the angles of elevation to the top of the pillar are respectively \(\theta\) and \(\phi\). If the height of the pillar is \(h\), find the distance between the two points.

15. From a point A on the ground, the angle of elevation to the top of a vertical pillar is 30°. After moving 20 meters toward the base of the pillar and reaching point B, the angle of elevation increases to 60°. Find the height of the pillar and the distance from point A to the pillar.

16. From a point on a horizontal line at the same level as the base of a chimney, a person walks 50 meters towards the chimney. As a result, the angle of elevation to the top of the chimney increases from 30° to 60°. Find the height of the chimney.

17. Let the smaller pillar be AB \(= x\) meters and the larger pillar be CD \(= 2x\) meters.
The midpoint of the base connection BD is O; the angles of elevation from O to the tops of the pillars are \(\angle\)AOB = \(\theta\) and \(\angle\)COD = 90\(^o-\theta\).
Since O is the midpoint of BD, we have BO = OD = \(\frac{120}{2}\) meters = 60 meters.

From \( \triangle \)ABO, we get:
\(\cfrac{AB}{BO} = \tan \theta\)
Or, \(\cfrac{x}{60} = \tan\theta\) --------(i)

From \( \triangle \)COD, we get:
\(\cfrac{CD}{OD} = \tan(90^o - \theta)\)
Or, \(\cfrac{2x}{60} = \cot\theta\) --------(ii)

Multiplying equations (i) and (ii), we get:
\(\cfrac{x}{60} \times \cfrac{2x}{60} = \tan\theta \times \cot\theta\)
Or, \(\cfrac{2x^2}{60 \times 60} = 1\)
Or, \(x^2 = \cfrac{60 \times \cancel{60}30}{\cancel{2}}\)
Or, \(x = 30\sqrt2\)

\(\therefore\) The height of the smaller pillar is \(30\sqrt2\) meters.
And the height of the larger pillar is \(30\sqrt2 \times 2\) meters \(= 60\sqrt2\) meters.
(Proved).

18. From a point on the roof of a five-story building, the angle of elevation to the top of a monument and the angle of depression to its base are 60° and 30° respectively. If the height of the building is 16 meters, what is the height of the monument?

19. A three-storey building has a flagpole of 3.6 meters mounted on its roof. From a point on the street, the angles of elevation to the top and bottom of the flagpole are 50° and 45°, respectively. Find the height of the building. [Assume: \(\tan 50^\circ = 1.2\)]

20. The deck of a ship is 10 meters above sea level. Standing on the deck, Apu observes the top of a lighthouse at an angle of elevation of 60°, and its base at an angle of depression of 30°. Find the distance from the ship to the lighthouse and the height of the lighthouse.

21. Due to a storm, a telegraph post bent at a point above the ground, causing its top end to touch the ground at a distance of \(4\sqrt{3}\) meters from its base and form an angle of 30° with the horizontal. Determine how high above the ground the post was bent, and what the total height of the post was.

22. From the roof of a five-story building that is 18 meters high, the angle of elevation to the top of a monument is 45°, and the angle of depression to its base is 60°. Find the height of the monument.

23. Two poles are placed 120 meters apart, and the height of one pole is double that of the other. If the angles of elevation to the tops of the two poles from the midpoint of the line joining their bases are complementary, find the height of the shorter pole.

24. A telegraph post got bent at a point above the ground due to a storm, and its top touches the ground at a distance of \(8\sqrt{3}\) meters from its base, forming a 30° angle with the horizontal. Calculate and write how high from the ground the post was bent, and what the total height of the post was.

25. From a point on the ground, the angle of elevation to the top of a mobile tower is 60°, and the distance from that point to the base of the tower is 10 meters. Find the height of the tower.

(a) 10 meters (b) 10\(\sqrt3\) meters (c) \(\frac{10}{\sqrt3}\) meters (d) 100 meters

26. The translation is: **Two vertical pillars** of heights \(h_1\) and \(h_2\) units stand on the same horizontal plane. From the foot of the pillar of height \(h_1\), the angle of elevation to the top of the other pillar (of height \(h_2\)) is 60°, and from the foot of the pillar of height \(h_2\), the angle of elevation to the top of the pillar of height \(h_1\) is 45°. Prove that: \(h_1^2 = 3h_2^2\)

27. From a point on the roof of a five-storey building, the angle of elevation to the top of a monument and the angle of depression to its base are complementary angles. If the height of the monument is four times the height of the building, find the measures of the elevation and depression angles.

28. A point is located \(h\) meters above a lake. From this point, the angle of elevation to a cloud is \(α\), and the angle of depression to its reflection in the lake is \(β\). Prove that the distance from the point to the cloud is \[ \cfrac{2h \sec α}{\tan β − \tan α} \]

29. The heights of two pillars are \(h_1\) meters and \(h_2\) meters respectively. If the angle of elevation from the base of the second pillar to the top of the first pillar is 60°, and the angle of elevation from the base of the first pillar to the top of the second pillar is 45°, show that \(h_1^2 = 3h_2^2\).

30. The heights of two pillars are 45 meters and 15 meters respectively. From the base of the second pillar, the angle of elevation to the top of the first pillar is 60°. Determine the angle of elevation to the top of the second pillar from the base of the first pillar.