1. 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.

2. 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.

3. 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.

4. 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.

5. 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?

6. 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.

7. 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.

8. 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.

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

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. 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) \(30°\)

12. 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.

13. 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?

14. 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?

15. 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."

16. 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.

17. 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.

18. 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.

19. From a point 50 meters away from the base of an incomplete pillar, the angle of elevation to its top is 30°. How much taller must the pillar be so that the angle of elevation to its new top from the same point becomes 45°?

20. 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.

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

22. 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

23. 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).

24. The heights of two towers are 180 meters and 60 meters, respectively. If the angle of elevation from the base of the second tower to the top of the first tower is 60°, then what is the angle of elevation from the base of the first tower to the top of the second tower?

25. 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.

26. 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°?

27. A 30-meter high pole and a pillar are located on the same horizontal plane. The foot of the pillar is seen from the top of the pole at a depression angle of 30°, and the top of the pillar is seen from the base of the pole at an elevation angle of 60°. What is the height of the pillar and the distance between the pole and the pillar?

28. 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.

29. 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.]

30. 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?