1. A 5-meter-high house is located on one side of a park. From both the roof and the base of the house, the base and the top of a palm tree on the opposite side of the park are seen at angles of depression and elevation of 30° and 60°, respectively. What is the height of the palm tree? What is the distance between the house and the palm tree?

2. From the roof of a 15-meter-high house located at one end of a park, the base and top of a brick kiln chimney at the other end of the park are seen at angles of depression and elevation of 30° and 60°, respectively. Determine the height of the brick kiln chimney and the distance between the kiln and the house.

3. From the roof of a 15-meter-high house situated at one end of a park, the foot and the top of a brick kiln chimney located at the other end of the park are seen at angles of depression and elevation of 30° and 60°, respectively. Find the height of the chimney and the distance between the brick kiln and the house.

4. 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.

5. From the top and the base of a vertical mountain, the angles of depression and elevation to the top of a 30-meter high pillar are respectively 30° and 60°. What is the height of the mountain?

6. Here is the English translation: > Two pillars of equal height are located directly opposite each other at points A and B on either side of a 120-meter wide road. From point C, on the line joining their bases, the angles of elevation to the tops of the pillars at A and B are 60° and 30°, respectively. Find the length of AC.

7. 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.

8. The heights of two pillars are 180 meters and 60 meters respectively. If the angle of elevation to the top of the second pillar from the base of the first pillar is 30°, then find the angle of elevation to the top of the first pillar from the base of the second pillar.

9. 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.

10. 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.

11. From the roof and the base of a 16-meter high building, the angles of elevation to the top of a temple are 45° and 60°, respectively. Find the height of the temple and its horizontal distance from the building. (The base of the building and the temple lie on the same horizontal plane.)

12. The heights of two pillars are 120 meters and 40 meters respectively. If the angle of elevation from the base of the second pillar to the top of the first pillar is 60°, then what is the angle of elevation from the base of the first pillar to the top of the second pillar?

13. 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).

14. The heights of two pillars are 180 meters and 60 meters, respectively. If the angle of elevation from the base of the second pillar to the top of the first pillar is 60°, determine the angle of depression from the base of the first pillar to the top of the second pillar.

15. The heights of two pillars are 180 meters and 60 meters. If the angle of elevation from the base of the second pillar to the top of the first pillar is 60°, what is the angle of elevation from the base of the first pillar to the top of the second pillar?

16. 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.

17. 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: \(h_1^2 = 3h_2^2\)

18. The heights of two pillars are 180 meters and 60 meters respectively. If the angle of elevation from the base of the second pillar to the top of the first pillar is 60°, find the angle of elevation from the base of the first pillar to the top of the second pillar.

19. 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 \(h_1^2 = 3h_2^2\).

20. The heights of two pillars are \(180\) meters and \(60\) meters, respectively. The angle of elevation from the base of the second pillar to the top of the first pillar is \(60°\). Determine the angle of elevation from the base of the first pillar to the top of the second pillar.

(a) \(90°\) (b) \(60°\) (c) \(45°\) (d) \(30°\)

21. The heights of two pillars are 180 meters and 60 meters, respectively. If the angle of elevation to the top of the first pillar from the base of the second pillar is 60°, then determine the angle of elevation to the top of the second pillar from the base of the first pillar.

22. The heights of two pillars are 180 meters and 60 meters, respectively. If the angle of elevation to the top of the first pillar from the base of the second pillar is 60°, determine the angle of elevation to the top of the second pillar from the base of the first pillar.

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. The English translation is: "The heights of two pillars are 180 meters and 60 meters respectively. If the angle of elevation from the base of the second pillar to the top of the first is 60°, determine the angle of elevation from the base of the first pillar to the top of the second."

25. The heights of two pillars are 180 meters and 60 meters, respectively. If the angle of elevation to the top of the first pillar from the base of the second pillar is 60°, calculate and write the angle of elevation to the top of the second pillar from the base of the first pillar.

26. A flag of 3.3 meters in length is mounted on the roof of a three-story building. From a certain point on the street, the angles of elevation to the top and bottom of the flagpole are 50° and 45°, respectively. Determine the height of the building. [tan 50° = 1.192]

27. 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?

28. In our neighborhood, there are two houses on opposite sides of the road facing each other. If the base of a ladder is placed 6 meters away from the foot of the wall of the first house and the ladder is leaned against the wall, it makes an angle of 30° with the horizontal. But if the same ladder is placed in the same position and leaned against the wall of the second house, it makes an angle of 60° with the horizontal. (i) Find the length of the ladder. (ii) Calculate how far the foot of the ladder is from the base of the second house. (iii) Find the width of the road. (iv) Determine the height at which the top of the ladder touches the second house.

29. If the angles of depression from the top of a lighthouse to the bases of the masts of two ships situated along the same straight line are 60° and 30° respectively, and the nearer ship is 150 meters away from the lighthouse, then calculate the distance of the farther ship from the lighthouse and the height of the lighthouse.

30. A vertical pole of height 126 decimeters is bent at some point above the ground, causing its upper part to tilt and touch the ground, forming a 30° angle with the horizontal surface. Calculate and write how high from the ground the pole was bent and how far from the base the tip of the pole touches the ground.