The document contains a comprehensive overview of trigonometric identities, relationships between different trigonometric functions, and numerous illustrative problems involving angles of elevation and depression related to various physical scenarios. It provides formulas for sine, cosine, tangent, and other trigonometric ratios, as well as methods for problem-solving using these identities. Numerous example problems are presented to demonstrate calculations for real-life applications such as tower heights, distances, and angles from points of observation.

2. 1)Sin
2)Cos
3)Tan
4)Cot
5)Sec
6)Cosec
Sin
cosec
sin 𝜃𝑋 𝑐𝑜𝑠𝑒𝑐𝜃 = 1
cos
cos 𝜃𝑋 𝑠𝑒𝑐𝜃 = 1
tan
cot
tan𝜃 𝑋 𝑐𝑜𝑡𝜃 = 1

3. SQUARE IDENTITIES
1) Sin2𝜽 + cos2𝜽 = 1
2) Sin2𝜽 = 1 - cos2𝜽
3) cos2𝜽 = 1 - sin2𝜽
4) sec2𝜽 - tan2𝜽 = 1
5) sec2𝜽 = 1 + tan2𝜽
6) sec2𝜽 - 1 = tan2𝜽
7) cosec2𝜽 - cot2𝜽 = 1
8) cosec2𝜽 = 1 + cot2𝜽
9) cosec2𝜽 - 1 = cot2𝜽
1) Sin2𝜽 =
1
cosec2𝜽
2) Sin2𝜽 = 1 - cos2𝜽
3) cos2𝜽 =
1
sec2𝜽
4) cos2𝜽 = 1 - sin2𝜽
5) tan2𝜽 =
1
cot2𝜽
6) tan2𝜽 =
Sin2𝜽
cos2𝜽
7) tan2𝜽 = sec2𝜽 – 1
8) cot2𝜽 =
1
tan2𝜽
9) cot2𝜽 =
cos2𝜽
sin2𝜽
10) cot2𝜽 = 𝐜𝐨sec2𝜽 – 1
11) sec2𝜽 =
1
cos2𝜽
12) sec2𝜽 = 1 + tan2𝜽
13) cosec2𝜽 =
1
sin2𝜽
14) cosec2𝜽 = 1 + cot2𝜽

4. Prove:

5. QUESTIONSBASED ON CROSS MULTIPLICATION

8. QUESTIONSBASED ON RATIONALIZATION

23. iv)
𝒕𝒂𝒏 𝑨
𝟏 − 𝒄𝒐𝒕 𝑨
+
𝒄𝒐𝒕 𝑨
𝟏 − 𝒕𝒂𝒏𝑨
= sec A . cosec A + 1
LHS :
𝒕𝒂𝒏 𝑨
𝟏 − 𝒄𝒐𝒕 𝑨
+
𝒄𝒐𝒕 𝑨
𝟏 − 𝒕𝒂𝒏𝑨
TAKING LCM
𝑼𝒔𝒊𝒏𝒈 𝒂𝟑 − 𝒃𝟑
= 𝐚 − 𝒃 𝒂𝟐 + 𝒂𝐛 + 𝒃𝟐
𝒔𝒊𝒏𝑨 − 𝐜𝐨𝐬 𝑨 𝐬𝐢𝐧𝟐
𝑨 + 𝐬𝐢𝐧 𝑨 𝐜𝐨𝐬𝑨 + 𝐜𝐨𝐬𝟐
𝑨
𝐬𝐢𝐧𝑨 𝐜𝐨𝐬 𝑨 𝐬𝐢𝐧𝑨 − 𝐜𝐨𝐬 𝑨
[𝒔𝒊𝒏𝟐 𝜽 + 𝒄𝒐𝒔𝟐 𝜽 = 𝟏]
sec A . cosec A + 1 LHS = RHS
RHS

24. iv)
𝒄𝒐𝒕 𝑨
𝟏 −𝒕𝒂𝒏 𝑨
+
𝒕𝒂𝒏𝑨
𝟏 −𝒄𝒐𝒕 𝑨
= 1 + tan A + cot A
LHS :
𝒄𝒐𝒕 𝑨
𝟏 −𝒕𝒂𝒏 𝑨
+
𝒕𝒂𝒏𝑨
𝟏 −𝒄𝒐𝒕 𝑨
TAKING LCM
𝑼𝒔𝒊𝒏𝒈 𝒂𝟑 − 𝒃𝟑
= 𝐚 − 𝒃 𝒂𝟐 + 𝒂𝐛 + 𝒃𝟐
𝒄𝒐𝒔𝑨 − 𝒔𝒊𝒏𝑨) 𝒄𝒐𝒔𝟐
𝑨 + 𝐬𝐢𝐧𝑨 𝐜𝐨𝐬𝑨 + 𝐬𝐢𝐧𝟐
𝑨
𝐬𝐢𝐧𝑨 𝐜𝐨𝐬 𝑨 𝒄𝒐𝒔𝑨 − 𝒔𝒊𝒏𝑨)

25. Prove that : ( 1 + cot A – cosec A) ( 1 + tan A + sec A) = 2

26. Prove That:

27. Prove That:

28. Prove That:

30. HINT FOR DRAWING THE DIAGRAM
1) DRAW VERTICAL LINE THAT WILL REPRESENT BUILDING , POLE , TOWER
, CLIFF , TREE , ETC.
2) MARK THE POSITION OF AN OBSERVER
3) DRAW LINE OF VISION AND HORIZONTAL LINE AND MARK THE ANGLE
OF ELEVATION OR ANGLE OF DEPRESSON
LINE OF VISION : The line which is drawn from the eyes of the
observer to the point being viewed on the object is known
as the line of sight.
HORIZONTAL LINE :- a base line on which observer is
standing
ANGLE OF ELEVATION :- the angle formed by the line of sight
and the horizontal plane for an object ABOVE the
horizontal. ( WHEN OBSERVER LOOKS FROM BOTTOM TO
TOP )
ANGLE OF DEPRESSION :- the angle formed by the line of
sight and the horizontal plane for an object BELOW the
horizontal. ( WHEN OBSERVER LOOKS FROM TOP TO
BOTTOM )

31. The angle of elevation of the top of a tower from a point on the
ground and at a distance of 160 m from its foot, is found to be
60o. Find the height of the tower.

32. From the top of a cliff 92 m high, the angle of depression of a
buoy is 20o. Calculate, to the nearest metre, the distance of the
buoy from the foot of the cliff.

33. The angle of elevation of the top of an unfinished tower at a point distance 80 m from
its base is 30o. How much higher must the tower be raised so that its angle of elevation
at the same point may be 60o?

34. A boy, 1.6 m tall, is 20 m away from a tower and observes the angle of elevation of
the top of the tower to be (i) 45o, (ii) 60o. Find the height of the tower in each case.

35. Find the height of a tree when it is found that on walking away from it 20
m, in a horizontal line through its base, the elevation of its top changes
from 60o to 30o.

36. Find the height of a building, when it is found that on walking
towards it 40 m in a horizontal line through its base the angular
elevation of its top changes from 30o to 45o.

37. A person standing on the bank of a river observes that the angle of
elevation of the top of a tree standing on the opposite bank is 60o. When he
moves 40 m away from the bank, he finds the angle of elevation to be 30o.
Find:
(i) the height of the tree, correct to 2 decimal places,
(ii) the width of the river.

38. A man standing on the bank of a river observes that the angle of elevation of a tree
on the opposite bank is 60o. When he moves 50 m away from the bank, he finds the
angle of elevation to be 30o. Calculate:
(i) the width of the river;
(ii) the height of the tree.

39. A man observes the angle of elevation of the top of a building to be 30o. He walks
towards it in a horizontal line through its base. On covering 60 m, the angle of elevation
changes to 60o. Find the height of the building correct to the nearest metre.

40. As observed from the top of a 80 m tall lighthouse, the angles of depression of two
ships, on the same side of a light house in a horizontal line with its base, are 30° and
40° respectively. Find the distance between the two ships. Give your answer corrected
to the nearest metre.

41. The angles of depression of two ships A and B as observed from the top of a light
house 60m high, are 60° and 45° respectively. If the two ships are on the opposite
sides of the light house, find the distance between the two ships. Give your answer
correct to the nearest whole number.

42. Two climbers are at points A and B on a vertical cliff face. To an observer C, 40m
from the foot of the cliff, on the level ground, A is at an elevation of 48o and B of 57o.
What is the distance between the climbers?

43. A man stands 9 m away from a flag-pole. He observes that angle of
elevation of the top of the pole is 28o and the angle of depression
of the bottom of the pole is 13o. Calculate the height of the pole.

44. Two pillars of equal heights stand on either side of a roadway, which is
150 m wide. At a point in the roadway between the pillars the elevations
of the tops of the pillars are 60o and 30o ; find the height of the pillars
and the position of the point.

45. The angle of elevation of the top of a tower is observed to be 60o. At a
point, 30 m vertically above the first point of observation, the elevation is
found to be 45o. Find:
(i) the height of the tower,
(ii) its horizontal distance from the points of observation.

46. Q12) The horizontal distance between two towers is 75 m and the angular
depression of the top of the first tower as seen from the top of the
second, which is 160 m high, is 45o. Find the height of the first tower.

47. Q15) From the top of a hill, the angles of depression of two consecutive
kilometer stones, due east, are found to be 30o and 45o respectively. Find
the distances of the two stones from the foot of the hill.

48. Q13) A 20 m high vertical pole and a vertical tower are on the same level ground in
such a way that the angle of elevation of the top of the tower, as seen from the foot of
the pole is 60o and the angle of elevation of the top of the pole, as seen from the foot of
the tower is 30o. Find:
(i) the height of the tower ;
(ii) the horizontal distance between the pole and the tower.

49. Q14) A vertical pole and a vertical tower are on the same level ground in such a way that
from the top of the pole, the angle of elevation of the top of the tower is 60o and the
angle of depression of the bottom of the tower is 30o. Find:
(i) the height of the tower, if the height of the pole is 20 m;
(ii) the height of the pole, if the height of the tower is 75 m.

50. Q15) From a point, 36 m above the surface of a lake, the angle of elevation of a bird is
observed to be 30o and the angle of depression of its image in the water of the lake is
observed to be 60o. Find the actual height of the bird above the surface of the lake.

51. Q19) An aeroplane, at an altitude of 250 m, observes the angles of depression of two
boats on the opposite banks of a river to be 45° and 60° respectively. Find the width of
the river. Write the answer correct to the nearest whole number.

52. Q20) The horizontal distance between two towers is 120 m. The angle of
elevation of the top and angle of depression of the bottom of the first tower as
observed from the top of the second tower is 30° and 24° respectively. Find the
height of the two towers. Give your answers. Give your answer correct to 3
significant figures.

53. A man on the top of the tower observes a car moving at a uniform speed coming
directly towards it. If it takes 12 mins for the angle of depression to change from
30 to 45 how soon after this will the car reach the tower

54. A man on the top of the tower observes a car moving at a uniform speed going
away from it. If it takes 2 mins for the angle of depression to change from 60 to
45 . Find the speed of the boat if the height of the tower is 150m.

55. An aeroplane flying horizontally 1 km above the ground and going away
from the observer is observed at an elevation of 60o. After 10 seconds,
its elevation is observed to be 30o; find the uniform speed of the
aeroplane in km per hour.

56. The upper part of a tree, broken over by the wind, makes an angle of
45o with the ground and the distance from the root to the point where the
top of the tree touches the ground is 15 m. What was the height of the
tree before it was broken?

57. At a point on level ground, the angle of elevation of a vertical tower
is found to be such that its tangent is . On walking 192 metres
towards the tower, the tangent of the angle is found to be .
Find the height of the tower.