Assignments For Class 10 Mathematics Applications Of Trigonometry

Assignments for Class 10 Mathematics Applications Of Trigonometry have been developed for Standard 10 students based on the latest syllabus and textbooks applicable in CBSE, NCERT and KVS schools. Parents and students can download the full collection of class assignments for class 10 Mathematics Applications Of Trigonometry from our website as we have provided all topic wise assignments free in PDF format which can be downloaded easily. Students are recommended to do these assignments daily by taking printouts and going through the questions and answers for Grade 10 Mathematics Applications Of Trigonometry. You should try to do these test assignments on a daily basis so that you are able to understand the concepts and details of each chapter in your Mathematics Applications Of Trigonometry book and get good marks in class 10 exams.

Question. If the angle of elevation of cloud from a point ‘h’ meters above a lake is α and the angle of depression of its reflection in the lake is β then the height of the cloud is=?   
A) h(tanα+tanβ)/(tanβ-tanα)
B) (tanα+tanβ)/(tanβ-tanα)
c) (tanα-tanβ)/(tanβ+tanα)
D) None of these

Question. A man on a top of tower observes a truck at an angle of depression α where tanα = 1/√5 and sees that it is moving towards the base og the tower. Ten minutes later, the angle of depression of the truck is found to be β where tan β = 5 , if the truck is moving at a uniform speed, determine how much more time it will take to reach the base of the tower   
A) 100 sec
B) 200 sec
C) 150 sec
D) 250 sec

Question. Two ships are sailing in the sea on either side of a lighthouse; The angles of depression of two ships as observed from the lighthouse are 0 60 and 0 45 respectively. If the distance between the ships is 200 (1+√3 /√3 )  meters, then the   
A) 150 m
B) 200m
C) 250 m
D) None of these

Question. The angle of elevation of the top of the tower standing on a horizontal plane from a point A is α. After walking a Distance ‘d’ towards the foot of the tower the angle of elevation is found to be β. Then height of the tower is?   
A) d / cotα. cotβ
B) d / cotα. cotβ
C) d / sinα. sinβ
D) NONE

Question. A round balloon of radius ‘a’ subtends an angle θ at the eye of the observer while the angle of elevation of its centre is Φ . Then the height of the center of the balloon.   
A) asinθ.cosec4
B) asinθ .sinΦ
C)  acosα4/2 .sinΦa ec Φ
D) None of these

Question. A ladder sets against a wall at an angle α to the horizontal. If the root is pulled away from the wall through a distance of ‘a’, so that is slides a distance ‘b’ down the wall making an angle β with the horizontal. Then  cosα – cosβ /sinβ – sinα   
A) (b/a)
B) (a/b)
C) (2b/a)
D) (2a/b)

Question. From an aero plane vertically above a staright horizontal road, the angles of depression of two consecutive milestones on opposite sides of the aero plane are observed to be α and β . The height of the aero plane above the road is   
A) tanα .tanβ/1+ tanα .tanβ
B) tanα .tanβ/ tanα +tanβ
C) tanα .tanβ/ tanα -tanβ
D) None of these

Question. The stations due south of a tower, which learns towards north are at distances ‘a’ and ‘b’ from its foot. If α and β be the elevations of the top of the tower from the situation, then its inclination ‘Θ ’ to the horizontal given by   
A) {bcotα-acotβ/ba}
B) {bcotα+acotβ/b-a}
C)  {bcotα-acotβ/b+a}
D) None of these

Question. The angle of elevation of the top of a tower from a point on the same level as the foot of the tower is α . On advancing ‘p’ meters towards the foot of the tower, the angle of elevation becomes β . The height ‘h’ of the tower is given by h=?   
A) p(tanα x tanβ)/(tanβ + tanα)
B) p(tanα x tanβ)/(tanβ + tanα)
c)  (tanα x tanβ)/(tanβ + tanα)
D) None of these

Question. A boy standing on a horizontal plane finds a bird flying at a distance of 100 m from him at an elevation of 0 30 . A girl standing on the roof of 20 meter high building finds the angle of elevation of the same bird to be 450. Both the boy and the girl are on opposite sides of the bird. The distance of the bird from the girl is   
A) 20.42 m
B) 42.42 m
C) 42.32 m
D) None of these

Question. From a window x meters high above the ground in a street, the angles of elevation and depression of the top and the foot of the other house on the opposite side of the street are α and β respectively. The height of the opposite house is =?   
A) x(1+ tanα.cotβ)
B) x(1- tanα.cotβ)
C) x(1+sinα.cotβ)
D) None of these

Question. The angle of elevation of a jet fighter from a point A on the ground is 0 60 . After a fight of 15 seconds, the angle of elevation changes to 300. If the jet is flying at a speed of 720 km/hr then constant height at which the jet is flying. ( Use 3 √1.732 )
A) 2598 m
B) 2600 m
C) 2500 m
D) 2550 m

Question. Two stations due south of a leaning tower which leans towards the north are at distance a and b from its foot. If α ,β be the elevations of the top of the tower from these stations, prove that its inclination Φ is given by b cotα -a cot β /b-a = ?   
A) cosΦ
B) sinΦ
C) cotΦ
D) tanΦ

CBSE Class 10 Mathematics Applications Of Trigonometry Assignment Set A

CBSE Class 10 Mathematics Applications Of Trigonometry Assignment Set B

CBSE Class 10 Mathematics Applications Of Trigonometry Assignment Set C

CBSE Class 10 Mathematics Applications Of Trigonometry Assignment Set E

CBSE Class 10 Mathematics Applications Of Trigonometry Assignment Set G

CBSE Class 10 Mathematics Applications Of Trigonometry Assignment Set I

CBSE Class 10 Mathematics Applications Of Trigonometry Assignment Set K

CBSE Class 10 Mathematics Applications Of Trigonometry Assignment Set L

CBSE Class 10 Mathematics Applications Of Trigonometry Assignment Set M

CBSE Class 10 Mathematics Applications Of Trigonometry Assignment Set N

CBSE Class 10 Mathematics Applications Of Trigonometry Assignment Set O

CBSE Class 10 Mathematics Applications Of Trigonometry Assignment Set P

CBSE Class 10 Mathematics Applications Of Trigonometry Assignment Set Q

CBSE Class 10 Mathematics Applications Of Trigonometry Assignment Set R

CBSE Class 10 Applications of Trigonometry (10)_1

CBSE Class 10 Applications of Trigonometry (11)_1

CBSE Class 10 Applications of Trigonometry (12)_1

CBSE Class 10 Applications of Trigonometry (13)_1

Some Applications of Trigonometry

Short Answer Type Questions :

Question. The rod AC of a TV disc antenna is fixed at right angles to the wall AB and a rod CD is supporting the disc as shown in the figure. If AC = 1.5 m long and CD = 3 m, find

(a) tan q
(b) sec q + cosec q
Solution. ΔACD is a right angled triangle.

Question. The angle of depression of the top and bottom of a tower as seen from the top of a 60 √3m high cliff are 45° and 60° respectively. Find the height of the tower. 
Solution. In ΔABC,

Question. The angle of depression from the top of a tower of a point A on the ground is 30°. On moving a distance of 20 m from the point A towards the foot of the tower to the point B, the angle of elevation of the top of the tower from the point B is 60°. Find the height of the tower and its distance from the point A. 
Solution. In right ΔDCB,

Question. A tower stands vertically on the ground. From a point on the ground, which is 15m away from the foot of the tower, the angle of elevation of the top of the tower is found to be 60°. Find the height of the tower.
Solution. In right ΔABC, tan 60° = h/15

Question. A circus artist is climbing a 20 m long rope, which is tightly stretched and tied from the top of a vertical pole to the ground. Find the height of the pole, if the angle made by the rope with the ground level is 30° (see figure given alongside). 
Solution. In the given figure, let AC be the rope and AB be the pole.

Question. A flag-staff stands on the top of a 12 m high tower. From a point on the ground, the angles of elevation of the top and bottom of the flag-staff are observed to be 45° and 30° respectively. Find the height of the flag-staff. 
Solution. 12(√3 – 1) m

Question. The two palm trees are of equal heights and are standing opposite to each other on either side of the river, which is 80 m wide. From a point O between them on the river, the angles of elevation of the top of the trees are 60° and 30°, respectively. Find the height of the trees and the distances of the point O from the trees.
Solution.

Question. The given figure shows a statue, 1.6 m tall, standing on the top of pedestal. From a point on the ground, the angle of elevation of the top of the statue is 60° and from the same point, the angle of elevation of the top of the pedestal is 45°. Find the height of the pedestal.

14. (0.8) (√3 + 1) m

Question. From the top of a 60 m high building, the angles of depression of the top and the bottom of a tower are 45° and 60° respectively. Find the height of the tower. [Take √3 = 1.73]

Solution. Let AB be 60 m high building and CD be the tower of height h.
∴ ∠ACE = 45° and ∠ADB = 60°
(using alternate angles)
Let BD = CE = x
BE = CD = h fi AE = 60 – h
In right-angled triangle ABD,

Height of the tower is 25.4 m.

Question. The angle of elevation of an aeroplane from a point on the ground is 60°. After a flight of 30 seconds the angle of elevation becomes 30°. If the aeroplane is flying at a constant height of 3000 3 m, find the speed of the aeroplane. 
Solution. In right-angled triangle OLA,

Question. From a point P on the ground, the angle of elevation of the top of a 10 m tall building is 30°. A flagstaff is fixed at the top of the building and the angle of elevation of the top of the flagstaff from the point P is 45°. Find the length of the flagstaff and the distance of building from the point P.
Solution. Let height of flagstaff be h and the distance of the building from the point P be x.

Question. From the top of a 7 m high building, the angle of elevation of the top of a cable tower is 60° and the angle of depression of its foot is 30°. Determine the height of the tower. 
Solution. In ΔADC,

Question. A man on the top of a vertical observation tower observes a car moving at a uniform speed coming directly towards him. If it takes 12 minutes for the angle of depression to change from 30° to 45°, how soon after this, will the car reach the observation tower? 
Solution. 16.39 mins.

Question. The angle of elevation of the top Q of a vertical tower PQ from a point X on the ground is 60°. From a point Y, 40 m vertically above X, the angle of elevation of the top Q of tower PQ is 45°. Find the height of the tower PQ and the distance PX. [Use √3 = 1.73]
Solution. In ΔQPX, PQ/PX = tan 60° ⇒ h/z = √3   
h/√3 = z

Question. From the top of a tower 50 m high, the angle of depression of the top of a pole is 45° and from the foot of the pole, the angle of elevation of the top of the tower is 60°. Find the height of the pole if the pole and tower stand on the same plane. 
Solution. Let the height of the pole is h.
In right ΔEDC,

Question. Two men on either side of a 75 m high building and in line with base of building observe the angles of elevation of the top of the building as 30° and 60°. Find the distance between the two men. (Use √3 = 1.73).
Solution. Let C and D be the positions of two men.
Let CB = y and BD = x

Question. A kite is attached to a string. Assuming that there is no slack in the string, find the height of the kite above the level of the ground, if the length of the string is 54 m and it makes an angle of 30° with the ground. 
Solution. 27 m

Question. The angles of depression of the top and bottom of a 50 m high building from the top of a tower are 45° and 60° respectively. Find the height of the tower and the horizontal distance between the tower and the building. [Use √3 = 1.73]
Solution. In ΔAED, y/x = tan 45° ⇒ y/x = 1

Question. A 7 m long flagstaff is fixed on the top of a tower standing on the horizontal plane. From a point on the ground, the angles of elevation of the top and bottom of the flagstaff are 60° and 45° respectively. Find the height of the tower correct to one place of decimal. [Use √3 = 1.73]
Solution. In ΔABC, AB/BC = tan 45°
h/x = 1 ⇒ h = x

Question. In the given figure, the angle of elevation of the top of a tower from a point C on the ground, which is 30 m away from the foot of the tower, is 30°. Find the height of the tower.
Solution. 10√3 m

Question. As observed from the top of a 60 m high lighthouse from the sea-level, the angles of depression of the two ships are 30° and 45°. If one ship is exactly behind the other on the same side of the lighthouse, find the distance between the two ships. [Use √3 = 1.732]
Solution. In ΔABP,

Question. The angles of elevation of the top of a tower from two points at a distance of 6 m and 13.5 m from the base of the tower and in the same straight line with it are complementary. Find the height of the tower. 
Solution. In ΔABC, tan θ = h/6

Question. Two ships are there in the sea on either side of a lighthouse in such a way that the ships and the lighthouse are in the same straight line. The angles of depression of two ships as observed from the top of the lighthouse are 60° and 45°. If the height of the lighthouse is 200 m, find the distance between the two ships. [Use √3 = 1.73]
Solution. In ΔABD,

Question. The ratio of the length of a vertical rod and the length of its shadow is 1 : √3 . Find the angle of elevation of the sun at that moment?
Solution. 30°

Question. The angles of depression of the top and bottom of a 8 m tall building from the top of a multi-storied building are 30° and 45°, respectively. Find the height of the multi-storied building and the distance between the two buildings.
Solution. Let AB be the multi-storied building of height h m and CD the building at a distance x m.

Question. In the given figure, the angle of elevation of the top of a tower AC from a point B on the ground is 60°. If the height of the tower is 20 m, find the distance of the point from the foot of the tower.

Solution. 20/√3

Question. As observed from the top of a lighthouse, 100 m high above sea level, the angles of depression of a ship, sailing directly towards it, changes from 30° to 60°. Find the distance travelled by the ship during the period of observation. [Use √3 = 1.73] 
Solution. In ΔABC, AB/BC = tan 60°

Question. The shadow of a tower standing on a level ground is found to be 30 m longer when the sun’s altitude is 30° than when it is 60°. Find the height of the tower.
Solution. Let AB be the tower of height h.
Let BC = x

Question. A 1.2 m tall girl spots a balloon moving with the wind in a horizontal line at a height 88.2 m from the ground. The angle of elevation of the balloon from the eyes of the girl at any instant is 60°. After sometime, the angle of elevation reduces 30°. Find the distance travelled by the balloon during the interval.

Solution.

Question. An aeroplane, when flying at a height of 4000 m from the ground passes vertically above another aeroplane at an instant when the angles of elevation of the two planes from the same point on the ground are 60° and 45° respectively. Find the vertical distance between the aeroplanes at that instant. [Take √3 = 1.73]
Solution. In ΔABC, AB/BC = tan 45° ⇒ 4000 − y/x = 1
⇒ x = (4000 – y) m

Question. An electrician has to repair an electric fault on a pole of height 4 m. He needs to reach a point 1.3 m below the top of the pole to undertake the repair work (see figure given along side). What should be the length of the ladder that he should use which, when inclined at an angle of 60° to the horizontal would enable him to reach the required position?

7. 5.4/√3m

Question. A man in a boat rowing away from a lighthouse 100 m high takes 2 minutes to change the angle of elevation of the top of the lighthouse from 60° to 30°. Find the speed of the boat in metres per minute. [Use √3 = 1.732]
Solution. Let BD = x m and CD = y m

Question. From a point on the ground, the angle of elevation of the top of a tower is observed to be 60°. From a point 40 m vertically above the first point of observation, the angle of elevation of the top of the tower is 30°. Find the height of the tower and its horizontal distance from the point of observation. 
Solution. Let h be the height of the tower and x be the horizontal distance from the point of observation.
∴ CB = ED = x and CE = BD = 40 m
In right ΔABC, 
tan 30° = AB/BC ⇒ 1/√3 = AB/x

Question. If the angles of elevation of the top of the candle from two coins distant ‘a’ cm and ‘b’ cm (a > b) from its base and in the same straight line from it are 30° and 60°, then find the height of the candle.

Solution.

Question. As observed from the top of a 100 m high lighthouse from the sea-level, the angles of depression of two ships are 30° and 45°. If one ship is exactly behind the other on the same side of the lighthouse, find the distance between the two ships. [Use √3 = 1.732]
Solution. Let AB be the tower and ships are at points C and D.

Question. The angle of elevation of a jet fighter from a point A on the ground is 60°. After a flight of 10 seconds, the angle of elevation changes to 30°. If the jet is flying at a speed of 900 km / hour, find the constant height at which the jet is flying. [Use √3 = 1.732]
Solution. 2165 m

Question. The angle of elevation of the top of a tower at a distance of 120 m from a point A on the ground is 45°. If the angle of elevation of the top of a flagstaff fixed at the top of the tower, at A is 60°, then find the height of the flagstaff. [Use √3 = 1.73]
Solution. Let BC = x be the tower and BD be the flagstaff of height h.
AC = 120 m, ∠BAC = 45° and ∠DAC = 60°

Question. As observed from the top of a 75 m highlight house above the sea level, the angles of depression of two ships are 30° and 45° respectively. If one ship is exactly behind the other on the same side of the lighthouse and in the same straight line, find the distance between the two ships. [Use √3 = 1.732]
Solution.

Question. A statue 1.6 m tall stands on the top of a pedestal. From a point on the ground, the angle of elevation of the top of the statue is 60° and from the same point the angle of elevation of the top of the pedestal is 45°. Find the height of the pedestal. [Use √3 = 1.73]
Solution. Let AD be 1.6 m tall statue, BD the pedestal and C the point of observation such that ∠ACB = 60° and ∠DCB = 45°
In right ΔABC,

Question. The angle of elevation of a cloud from a point 60 m above a lake is 30° and the angle of depression of the reflection of cloud in the lake is 60°. Find the height of the cloud.
Solution. In right ΔAEC,

Question. A round balloon of radius r subtends an angle a at the eye of the observer while the angle of elevation of its centre is b. Prove that the height of the centre of the balloon is r sin β cosec α/2 .
Solution. According to the given statement, the diagram will be as shown:

Question. From a point on a bridge across a river, the angles of depression of the banks on opposite sides of the river are 30° and 45°, respectively. If the bridge is at a height of 10 m from the banks, then find the width of the river. [Use √3 = 1.73]
Solution. Let BC be the river and AD = 10 m be the height of bridge,

Question. The angle of elevation of the top of a building from the foot of a tower is 30°. The angle of elevation of the top of the tower from the foot of the building is 60°. If the tower is 60 m high, find the height of the building.
Solution. Let AB be the building and CD be the tower of height 60 m.
In right ΔBCD,

Question. Two poles of equal heights are standing opposite to each other on either side of the road, which is 80 m wide. From a point between them on the road, the angles of elevation of the top of the poles are 60° and 30° respectively. Find the height of the poles and the distances of the point from the poles.
Solution. Let AB and CD be the two poles of equal height standing on the opposite sides of the road BD.
∴ AB = CD
From figure in right ∆ABE

Question. At the foot of a mountain, the angle of elevation of its summit is 45°. After ascending 1 km towards the mountain up an incline of 30°, the elevation changes to 60°. Find the height of the mountain. [Use √3 = 1.732] 
Solution. 1.366 km

Question. The shadow of a tower standing on a level ground is found to be 40 m longer when the Sun’s altitude is 30° than when it was 60°. Find the height of the tower. [Given √3 = 1.732]
Solution. Let AB be a tower of height h m, and BC its shadow when sun’s altitude is 60° and BD also its shadow when sun’s altitude is 30°.

Question. A verticle tower stands on a horizontal plane and is surmounted by a vertical flag-staff of height 6 m. At a point on the plane, the angle of elevation of the bottom and top of the flag-staff are 30° and 45° respectively. Find the height of the tower. [Take √3 = 1.73] 
Solution. Let AD be a flagstaff of height 6 m, CD = h be the tower and B be the point of observation.

Question. From a point on the ground, the angles of elevation of the bottom and the top of a tower fixed at the top of a 20 m high building are 45° and 60° respectively. Find the height of the tower. 
Solution. 14.64 m

Question. The angles of elevation and depression of the top and bottom of a lighthouse from the top of a 60 m high building are 30° and 60° respectively. Find
(i) the difference between the heights of the lighthouse and the building.
(ii) the distance between the lighthouse and the building. 
Solution. In right ΔABD,

Question. The angles of depression of the top and bottom of a building 50 metres high as observed from the top of a tower are 30° and 60°, respectively. Find the height of the tower and also the horizontal distance between the building and the tower.
Solution. In right ΔBTP,

To measures height and distance of objects
Angle of elevation : The angle which the line of sight makes with the horizontal line
through O in the upper direction is called the angle of elevation of P, as seen from O.

Line of sight : When an observer looks from a point O at an object P, then the line OP is called the line of sight.
Angle of depression : the angle which the line of sight makes with the horizontal line through O in the lower direction is called the angle of depression of P, as seen from O.

Altitude of the sun : The altitude of the sun is simply the angle of elevation of the sun. 

Question. If the angles of elevation of the top of a tower from two points distant a and b (a > b) from its foot and in the same straight line from it are respectively 30o and 60p, then find the eight of the tower.   
Solution. Let the height of tower be h.

Question. On a horizontal plane there is a vertical tower with a flag pole on the top of the tower. At a point 9 metres away from the foot of the tower the angle of elevation of the top and bottom of the flag pole are 60° and 30° respectively. Find the height of the tower and the flag pole mounted on it.   
Solution. Let us suppose that AB be the tower and BC be flagpole
Let us suppose that O be the point of observation. Then, OA = 9m
According to question it is given that

Hence, height of the tower is 5.196m and the height of the flagpole is 10.392 m

Question. A pole casts a shadow of length 2√3 on the ground, When the Sun’s elevation is 60° . Find the height of the pole. 
Solution. 

AB = 2√3 BC is the height of pole.

Therefore, height of the pole = 6 m.

Question. A vertical tower of height 90 m stands on the ground. The angle of elevation of the top of the tower as observed from a point on the ground is 60o. Find the distance of the point from the foot of the tower.   
Solution.

Question. In figure, a tower AB is 20 m high and BC, its shadow on the ground, is 20√3 m long Find the Sun’s altitude.   

Solution. Let the tower AB = 20m and shadow BC = 20√3m   

Question. A pole 5 m high is fixed on the top of a tower. The angle of elevation of the top of pole observed from a point A on the ground is 60o and the angle of depression of the point A from  the top of the tower is 45o. Find the height of the tower. (Take √3 = 1.732 )   
Solution. 

given: the height of the pole (CB) = 5m
let ‘h’ be the height of the tower, then;
In ΔABD  ,

Question. An observer 1.5 m tall is 28.5 m away from a tower. The angle of elevation of the top of the tower from her eyes is 45°. What is the height of the tower?   
Solution. Let AB be the tower of height h and CD be the observer of height 1.5 m at a distance of 28.5 m from the tower AB.
In ΔAED we have

Question. The pilot of an aircraft flying horizontally at a speed of 1200 km/hr. observes that the angle of depression of a point on the ground changes from 30° to 45° in 15 seconds. Find the height at which the aircraft is flying. 
Solution. Distance covered in 15 seconds = AB

Question. A ladder 15 m long just reaches the top of a vertical wall. If the ladder makes an angle of 60° with the wall, then find the height of the wall. 
Solution.

Question. A man standing on the deck of a ship, which is 10 m above water level. He observes the angle of elevation of the top of a hill as 60° and the angle of depression of the base of the hill as 30°. Calculate the distance of the hill from the ship and the height of the hill.    20
Solution. Suppose the man is standing on the deck of a ship at point A and let CD be the hill. It is given that the angle of depression of the base C of the hill CD observed from A is 30°
and the angle of elevation of the top D of the hill CD observed from A is 60°

Hence, the distance of the hill from the ship is 10 metres and the height of the hill is 40 metres.

Question. A tree standing on a horizontal plane is leaning towards east. At two points situated at distances a and b exactly due west on it, the angles of elevation of the top are respectively and . Prove that the height of the top from the ground is (b – a) tan α tan β/tan α- tan β . 
Solution.

Let OP be the tree and A, B be two points such that = OA = a and OB = b

Question. If the height and length of shadow of a man are the same, then find the angle of elevation of the Sun.   
Solution. 

Let BC be the height of man and AB be the shadow of the man.

Question. As observed from the top of a 150 m tall light house, the angles of depression of two ships approaching it are 30° and 45°. If one ship is directly behind the other, find the distance between the two ships.   
Solution.

Question. On the same side of a tower, two objects are located. When observed from the top of the tower, their angles of depression are 45° and 60°. If the height of the tower is 150 m, find the distance between the objects.   
Solution.

Let AB be the tower of height 150 m and two objects are located when top of tower are observed, makes an
angle of depression from the top and bottom of tower are 45° and 60°

Question. At the foot of a mountain the elevation of its summit is 45o. After ascending 1000 m towards the mountain up a slope of 30o inclination, the elevation is found to be 60^o.  Find the height of the mountain.   
Solution. Let AB be the mountain of height h m and C be its foot.

Question. The length of shadow of a tower on the plane ground is √3 times the height of the tower. Find the angle of elevation of the sun. 
Solution. According to the question, the length of shadow of the tower on the plane ground is √3 times the height of the tower.

Question. The angle of elevation of the top of a tower at a distance of 120 m from a point A on the ground is 45°. If the angle of elevation of the top of a flagstaff fixed at the top of the tower, at A is 60o, then find the height of the flagstaff. [Use √3 = 1.73].   
Solution. Height of flagstaff = CD = h m
Height of tower = BD = x m

Question. The angle of elevation of the top of a building from the foot of the tower is 30o and the angle of elevation of the top of the tower from the foot of the building is 45o. If the tower is 30 m high, find the height of the building.   
Solution. Let the height of the building be CD = h m. and distance between tower and building
be, BD = x m.
and height of the tower be AB = 30m

Question. The horizontal distance between two trees of different heights is 60 m. The angle of depression of the top of the first tree, when seen from the top of the second tree is 45°. If the height of the second tree is 80 m, find the height of the first tree. 
Solution.

Let height of first tree = h m
Height of second tree = 80 m
∴ AB = (80 – h)m
Distance between both trees = 60 m

Question. A person standing on the bank of a river, observes that the angle of elevation of the top of the tree standing on the opposite bank is 60o. When he retreats 20 m from the bank, he finds the angle of elevation to be 30^o. Find the height of the tree and the breadth of the river. 
Solution. Let the height be ‘h’ m and breadth of river be ‘b’ m. 

Question. If a pole 6m high throws shadow of 2 m, then find the angle of elevation of the sun.     
Solution.

given,
height of the pole = 6m
Let AB is pole and BC is its shadow

Question. In fig. AB is 6 m high pole and CD is a ladder inclined at an angle of 60 to the horizontal and reaches up to a point of pole. If AD = 2.54 m, find the length of the ladder, (use √3 = 1.73) 

Solution. 

Question. A kite is flying at a height of 60 m above the ground. The string attached to the kite is temporarily tied to a point on the ground. The inclination of the string with the ground is 60^o. Find the length of the string assuming that there is no slack in the string.   
Solution. Let A be the kite and CA be the string attached to the kite such that its one end is tied
to a point C on the ground. The inclination of the string CA with the ground is 60^o.
In ΔABC , we are given that ∠C = 60° and perpendicular AB = 60 m and we have
to find hypotenuse AC. So, we use the trigonometric ratio involving perpendicular and hypotenuse.

Question. A person observed the angle of elevation of the top of a tower is 30o. He walked 50 m towards the foot of the tower along level ground and found the angle of elevation of the top of the tower as 60o. Find the height of the tower. 
Solution.

Question. The elevation of a tower at a station A due north of is α and at a station B due west of A is β  prove that the height of the tower is AB sin α sinβ /√sin2 α – sin2 β .
Solution. Let OP be the tower and let A be a point due north of the tower OP and let B be the point due west of A. Such that ∠OAP = α and ∠OBP = β . Let h be the height of the tower.
In right-angled triangle OAP and OBP, we have

Question. A kite is flying at a height of 30 m from the ground. The length of the string from kite to the ground is 60 m. Assuming that there is no slack in the string, find the angle of elevation of the kite at the ground.   
Solution.

Question. From an aeroplane vertically above a straight horizontal plane, the angles of depression of two consecutive kilometer stones on the opposite sides of the aeroplane are found to be α and . Show that the height of the aeroplane is tan α tan β /tan α + tan β .   
Solution. 

Let us suppose that aeroplane is at A
Suppose B and C are two consecutive kilometre stones such that

Question. The horizontal distance between two poles is 15 m. The angle of depression of the top of first pole as seen from the top of second pole is 30o. If the height of the first pole is 24 m, find the height of the hsecond pole.[Use √3 = 1.732]   
Solution.

Question. A ladder, leaning against a wall, makes an angle of 60° with the horizontal.If the foot of the ladder is 2.5 m away from the wall, then find the length of the ladder. 
Solution.

Question. A man standing on the deck of a ship, which is 10 m above water level, observes the angle of elevation of the top of a hill as 60° and angle of depression of the base of the hill as 30°. Find the distance of the hill from the ship and height of the hill.   
Solution.

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