#### Chapters

Chapter 2: Polynomials

Chapter 3: Pair of Linear Equations in Two Variables

Chapter 4: Quadratic Equations

Chapter 5: Arithmetic Progression

Chapter 6: Co-Ordinate Geometry

Chapter 7: Triangles

Chapter 8: Circles

Chapter 9: Constructions

Chapter 10: Trigonometric Ratios

Chapter 11: Trigonometric Identities

Chapter 12: Trigonometry

Chapter 13: Areas Related to Circles

Chapter 14: Surface Areas and Volumes

Chapter 15: Statistics

Chapter 16: Probability

## Chapter 12: Trigonometry

#### RD Sharma solutions for Class 10 Mathematics Chapter 12 Trigonometry Exercise 12.1, 12.5 [Pages 29 - 35]

A tower stands vertically on the ground. From a point on the ground 20 m away from the foot of the tower, the angle of elevation of the top of the tower is 60°. What is the height of the tower?

A tower stands vertically on the ground. From a point on the ground, 20 m away from the foot of the tower, the angle of elevation of the top of the tower is 600. What is the height of the tower?

A ladder is placed along a wall of a house such that its upper end is touching the top of the wall. The foot of the ladder is 2 m away from the wall and the ladder is making an angle of 60° with the level of the ground. Determine the height of the wall.

An electric pole is 10 m high. A steel wire tied to the top of the pole is affixed at a point on the ground to keep the pole upright. If the wire makes an angle of 45° with the horizontal through the foot of the pole, find the length of the wire.

A kit is flying at a height of 75 metres from the ground level, attached to a string inclined at 60 to the horizontal. Find the length of the string to the nearest metre.

The length of a string between a kite and a point on the ground is 90 meters. If the string makes an angle O with the ground level such that tan O = 15/8, how high is the kite? Assume that there is no slack in the string.

A vertical tower stands on a horizontal plane and is surmounted by a vertical flag-staff. At a point on the plane 70 metres away from the tower, an observer notices that the angles of elevation of the top and the bottom of the flagstaff are respectively 60° and 45°. Find the height of the flag-staff and that of the tower.

A vertically straight tree, 15 m high, is broken by the wind in such a way that its top just touches the ground and makes an angle of 60° with the ground. At what height from the ground did the tree break?

A vertical tower stands on a horizontal plane and is surmounted by a vertical flag-staff of height 5 meters. At a point on the plane, the angles of elevation of the bottom and the top of the flag-staff are respectively 300 and 600. Find the height of the tower.

A person observed the angle of elevation of the top of a tower as 30°. 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 60°. Find the height of the tower.

The shadow of a tower, when the angle of elevation of the sun is 45°, is found to be 10 m. longer than when it was 600. Find the height of the tower.

A parachutist is descending vertically and makes angles of elevation of 45° and 60° at two observing points 100 m apart from each other on the left side of himself. Find the maximum height from which he falls and the distance of the point where he falls on the ground form the just observation point.

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.

The angle of elevation of a tower from a point on the same level as the foot of the tower is 30°. On advancing 150 metres towards the foot of the tower, the angle of elevation of the tower becomes 60°. Show that the height of the tower is 129.9 metres (Use `sqrt3 = 1.732`)

The angle of elevation of the top of a tower as observed form a point in a horizontal plane through the foot of the tower is 32°. When the observer moves towards the tower a distance of 100 m, he finds the angle of elevation of the top to be 63°. Find the height of the tower and the distance of the first position from the tower. [Take tan 32° = 0.6248 and tan 63° = 1.9626]

The angle of elevation of the top of a tower from a point A on the ground is 30°. Moving a distance of 20metres towards the foot of the tower to a point B the angle of elevation increases to 60°. Find the height of the tower & the distance of the tower from the point A.

From the top of a building, 15 m high the angle of elevation of the top of a tower is found to be 30°. From the bottom of the same building, the angle of elevation of the top of the tower is found to be 60°. Find the height of the tower and the distance between the tower and building.

On a horizontal plane, there is a vertical tower with a flagpole on the top of the tower. At a point 9 meters away from the foot of the tower the angle of elevation of the top and bottom of the flagpole are 60° and 30° respectively. Find the height of the tower and the flagpole mounted on it.

A tree breaks due to storm and the broken part bends so that the top of the tree touches the ground making an angle 30 ° with it. The distance between the foot of the tree to the point where the top touches the ground is 8 m. Find the height of the tree.

From a point *P* on the ground the angle of elevation of a 10 m tall building is 30°. A flag is hoisted at the top of the building and the angle of elevation of the top of the flag-staff from *P* is 45°. Find the length of the flag-staff and the distance of the building from the point P. (Take `sqrt3` = 1.732)

A 1.6 m tall girl stands at a distance of 3.2 m from a lamp-post and casts a shadow of 4.8 m on the ground. Find the height of the lamp-post by using (i) trigonometric ratios (ii) property of similar triangles.

A 1.5 m tall boy is standing at some distance from a 30 m tall building. The angle of elevation from his eyes to the top of the building increases from 30° to 60° as he walks towards the building. Find the distance he walked towards the building

The shadow of a tower standing on a level ground is found to be 40 m longer when Sun’s altitude is 30° than when it was 60°. Find the height of the tower

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

The angles of depression of the top and bottom of 8 m tall building from the top of a multistoried building are 30° and 45° respectively. Find the height of the multistoried building and the distance between the two buildings.

A statue, 1.6 m tall, stands on a top of pedestal, from a point on the ground, the angle of elevation of the top of 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.

A T.V. Tower stands vertically on a bank of a river. From a point on the other bank directly opposite the tower, the angle of elevation of the top of the tower is 60°. From a point 20 m away this point on the same bank, the angle of elevation of the top of the tower is 30°. Find the height of the tower and the width of the river.

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 45°. Determine the height of the tower.

As observed from the top of a 75 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.

The angle of elevation of the top of a building from the foot of the tower is 30° and the angle of elevation of the top of the tower from the foot of the building is 60°. If the tower is 50 m high, find the height of the building

From a point on a bridge across a river, the angles of depression of the banks on opposite side of the river are 30° and 45° respectively. If the bridge is at the height of 30 m from the banks, find the width of the river.

Two poles of equal heights are standing opposite each other an 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 30º, respectively. Find the height of poles and the distance of the point from the poles.

A man sitting at a height of 20 m on a tall tree on a small island in the middle of a river observes two poles directly opposite to each other on the two banks of the river and in line with the foot of the tree. If the angles of depression of the feet of the poles from a point at which the man is sitting on the tree on either side of the river are 60° and 30°respectively. Find the width of the river.

A vertical tower stands on a horizontal plane and is surmounted by a flagstaff of height 7m. At a point on the plane, the angle of elevation of the bottom of the flagstaff is 30º and that of the top of the flagstaff is 45º. Find the height of the tower.

The length of the shadow of a tower standing on the level plane is found to 2x meter longer when the sun's altitude is 30° than when it was 45°. Prove that the height of the tower is `x(sqrt3 + 1)` meters.

A tree breaks due to the storm and the broken part bends so that the top of the tree touches the ground making an angle of 30° with the ground. The distance from the foot of the tree to the point where the top touches the ground is 10 metres. Find the height of the tree.

A balloon is connected to a meteorological ground station by a cable of length 215 m inclined at 600 to the horizontal. Determine the height of the balloon from the ground. Assume that there is no slack in the cable.

Two men on either side of the cliff 80 m high observe the angles of an elevation of the top of the cliff to be 30° and 60° respectively. Find the distance between the two men.

Find the angle of elevation of the sum (sun's altitude) when the length of the shadow of a vertical pole is equal to its height.

A fire in a building B is reported on the telephone to two fire stations P and Q, 20 km apart from each other on a straight road. P observes that the fire is at an angle of 60° to the road and Q observes that it is at an angle of 45° to the road. Which station should send its team and how much will this team have to travel?

A man on the deck of a ship is 10 m above the water level. He observes that the angle of elevation of the top of a cliff is 45° and the angle of depression of the base is 300. Calculate the distance of the cliff from the ship and the height of the cliff.

A man standing on the deck of a ship, which is 8 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.

There are two temples, one on each bank of a river, just opposite to each other. One temple is 50 m high. From the top of this temple, the angles of depression of the top and the foot of the other temple are 30° and 60° respectively. Find the width of the river and the height of the other temple.

The angle of elevation of an aeroplane from a point on the ground is 45°. After a flight of 15 seconds, the elevation changes to 30°. If the aeroplane is flying at a height of 3000 metres, find the speed of the aeroplane.

An aeroplane flying horizontally 1 km above the ground is observed at an elevation of 60°. After 10 seconds, its elevation is observed to be 30°. Find the speed of the aeroplane in km/hr.

From the top of a 50 m high tower, the angles of depression of the top and bottom of a pole are observed to be 45° and 60° respectively. Find the height of the pole.

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.

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

The angle of elevation of the top of a vertical tower *PQ* from a point *X *on the ground is 60°. At a point *Y*, 40 m vertically above *X*, the angle of elevation of the top is 45°. Calculate the height of the tower.

The angle of elevation of a stationary cloud from a point 2500 m above a lake is 15° and the angle of depression of its reflection in the lake is 45°. What is the height of the cloud above the lake level? (Use tan 15° = 0.268)

If the angle of elevation of a cloud from a point h meters above a lake is a and the angle of depression of its reflection in the lake be b, prove that the distance of the cloud from the point of observation is `(2h sec alpha)/(tan beta - tan alpha)`

From an aeroplane vertically above a straight horizontal road, the angles of depression of two consecutive milestones on opposite sides of the aeroplane are observed to be α and β. Show that the height in miles of the aeroplane above the road is given by `(tan alpha tan beta)/(tan alpha + tan beta)`

PQ is a post of given height a, and AB is a tower at some distance. If α and β are the angles of elevation of B, the top of the tower, at P and Q respectively. Find the height of the tower and its distance from the post.

A ladder rests against a wall at an angle α to the horizontal. Its foot is pulled away from the wall through a distance a so that it slides a distance b down the wall making an angle β with the horizontal. Show that `a/b = (cos alpha - cos beta)/(sin beta - sin alpha)`

A tower subtends an angle 𝛼 at a point A in the plane of its base and the angle if depression of the foot of the tower at a point b metres just above A is β. Prove that the height of the tower is b tan α cot β

An observer, 1.5 m tall, is 28.5 m away from a tower 30 m high. Determine the angle of elevation of the top of the tower from his eye.

A carpenter makes stools for electricians with a square top of side 0.5 m and at a height of 1.5 m above the ground. Also, each leg is inclined at an angle of 60° to the ground. Find the length of each leg and also the lengths of two steps to be put at equal distances.

A boy is standing on the ground and flying a kite with 100 m of string at an elevation of 30°. Another boy is sanding on the roof of a 10 m high building and is flying his kite at an elevation of 45°. Both the boys are on opposite sides of both the kites. Find the length of the string that the second boy must have so that the two kites meet.

The angle of elevation of the top of a hill at the foot of a tower is 60° and the angle of elevation of the top of the tower from the foot of the hill is 30°. If height of the tower is 50 m, find the height of the hill.

Two boats approach a lighthouse in mid-sea from opposite directions. The angles of elevation of the top of the lighthouse from two boats are 30° and 45° respectively. If the distance between two boats is 100 m, find the height of the lighthouse.

From the top of a building AB, 60 m high, the angles of depression of the top and bottom of a vertical lamp post CD are observed to be 30° and 60° respectively. Find

1) the horizontal distance between AB and CD

2) the height of the lamp post.

3) the difference between the heights of the building and the lamp post.

From the top of a lighthouse, the angles of depression of two ships on the opposite sides of it are observed to be a and 3. If the height of the lighthouse be h meters and the line joining the ships passes through the foot of the lighthouse, show that the distance

`(h(tan alpha + tan beta))/(tan alpha tan beta)` meters

A straight highway leads to the foot of a tower of height 50 m. From the top of the tower, the angles of depression of two cars standing on the highway are 30° and 60° respectively. What is the distance the two cars and how far is each car from the tower?

The angles of elevation of the top of a rock from the top and foot of a 100 m high tower are respectively 30° and 45°. Find the height of the rock.

An observed from the top of a 150 m tall lighthouse, 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.

A flag-staff stands on the top of a 5 m high tower. From a point on the ground, the angle of elevation of the top of the flag-staff is 60° and from the same point, the angle of elevation of the top of the tower is 45°. Find the height of the flag-staff.

The angles of elevation of the top of a tower from two points at a distance of 4 m and 9 m. from the base of the tower and in the same straight line with it are complementary. Prove that the height of the tower is 6 m.

The angles of depression of two ships from the top of a lighthouse and on the same side of it are found to be 45° and 30° respectively. If the ships are 200 m apart, find the height of the lighthouse.

From the top of a light house, the angles of depression of two ships on the opposite sides of it are observed to be α and β. If the height of the light house be h metres and the line joining the ships passes through the foot of the light house, show that the distance between the ship is

`(h (tan ∝+tan ß))/ (tan ∝+tan ∝)`

From the top of the tower * h *metre high , the angles of depression of two objects , which are in the line with the foot of the tower are ∝ and ß (ß> ∝ ) cts .

A window of a house is *h* metre above the ground . From the window , the angles of elevation and depression of the top and bottom of another house situated on the opposite side of the lane are found to be` ∝ ∝`and `ß ß `respectively. Prove that the height of the house is `h(1+tan ∝ tan ß )` metres.

The lower window of a house is at a height of 2 m above the ground and its upper window is 4 m vertically above the lower window . At certain instant the angles of elevation of a balloon from these windows are observed to be 60^{0} and 30^{0 }respectively. Find the height of the balloon above the ground.

#### RD Sharma solutions for Class 10 Mathematics Chapter 12 Trigonometry [Pages 40 - 41]

The height of a tower is 10 m. What is the length of its shadow when Sun's altitude is 45°?

If the ratio of the height of a tower and the length of its shadow is `sqrt3:1`, what is the angle of elevation of the Sun?

What is the angle of elevation of the Sun when the length of the shadow of a vetical pole is equal to its height?

From a point on the ground, 20 m away from the foot of a vertical tower, the angle elevation of the top of the tower is 60°, What is the height of the tower?

If the angles of elevation of the top of a tower from two points at a distance of 4 m and 9 m from the base of the tower in the same straight line with it are complementary, find the height of the tower.

In the following figure, what are the angles of depression from the observing position O_{1} and O_{2}of the object at A?

The tops of two towers of height *x* and *y*, standing on level ground, subtend angles of 30º and 60º respectively at the centre of the line joining their feet, then find *x* : *y*.

The angle of elevation of the top of a tower at a point on the ground is 30º. What will be the angle of elevation, if the height of the tower is tripled?

*AB *is a pole of height 6 m standing at a point *B *and *CD *is a ladder inclined at angle of 60^{0 }to the horizontal and reaches upto a point D of pole . If *AD *= 2.54 m , find the length of the ladder.

An observer , 1.7 m tall , is` 20 sqrt3` m away from a tower . The angle of elevation from the eye of an observer to the top of tower is 30^{0 }. Find the height of the tower.

An observer, 1.5 m tall, is 28.5 m away from a 30 m high tower. Determine the angle of elevation of the top of the tower from the eye of the observer.

#### RD Sharma solutions for Class 10 Mathematics Chapter 12 Trigonometry [Pages 41 - 44]

The ratio of the length of a rod and its shadow is `1 : sqrt3`. The angle of elevation of the sum is

30°

45°

60°

*90*°

If the angle of elevation of a tower from a distance of 100 metres from its foot is 60°, then the height of the tower is

100\[\sqrt{3}\]

\[\frac{100}{\sqrt{3}} m\]

\[50 \sqrt{3}\]

\[\frac{200}{\sqrt{3}} m\]

If the altitude of the sum is at 60°, then the height of the vertical tower that will cast a shadow of length 30 m is

\[10\sqrt{3}\]

15 m

\[\frac{30}{\sqrt{3}} m\]

\[15\sqrt{2} m\]

If the angles of elevation of a tower from two points distant a and b (a>b) from its foot and in the same straight line from it are 30° and 60°, then the height of the tower is

\[\sqrt{a + b}\]

\[\sqrt{ab}\]

\[\sqrt{a - b}\]

\[\sqrt{\frac{a}{b}}\]

If the angles of elevation of the top of a tower from two points distant a and b from the base and in the same straight line with it are complementary, then the height of the tower is

ab

\[\sqrt{ab}\]

\[\frac{a}{b}\]

\[\sqrt{\frac{a}{b}}\]

From a light house the angles of depression of two ships on opposite sides of the light house are observed to be 30° and 45°. If the height of the light house is h metres, the distance between the ships is

\[\left( \sqrt{3} + 1 \right) \text{ h metres }\]

\[\left( \sqrt{3} - 1 \right) \text{ h metres }\]

\[\sqrt{3} \text{ h metres }\]

\[1 + \left( 1 + \frac{1}{\sqrt{3}} \right) \text{ h metres }\]

The angle of elevation of the top of a 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. The height of the tower is

\[\frac{d}{cot \alpha + cot \beta}\]

\[\frac{d}{cot \alpha + cot \beta}\]

\[\frac{d}{\tan \beta - \tan \alpha}\]

\[\frac{d}{\tan \beta - \tan \alpha}\]

The tops of two poles of height 20 m and 14 m are connected by a wire. If the wire makes an angle of 30° with horizontal, then the length of the wire is

12 m

10 m

8 m

6 m

From the top of a cliff 25 m high the angle of elevation of a tower is found to be equal to the angle of depression of the foot of the tower. The height of the tower is

25 m

50 m

75 m

100 m

The angles of depression of two ships from the top of a light house are 45° and 30° towards east. If the ships are 100 m apart. the height of the light house is

\[\frac{50}{\sqrt{3 + 1}} m\]

\[\frac{50}{\sqrt{3 - 1}} m\]

\[50 \left( \sqrt{3} - 1 \right) m\]

\[50 \left( \sqrt{3} + 1 \right) m\]

If the angle of elevation of a cloud from a point 200 m above a lake is 30° and the angle of depression of its reflection in the lake is 60°, then the height of the cloud above the lake is

200 m

500 m

30 m

400 m

The height of a tower is 100 m. When the angle of elevation of the sun changes from 30° to 45°, the shadow of the tower becomes x metres less. The value of x is

100 m

\[100\sqrt{3} m\]

\[100\left( \sqrt{3} - 1 \right) m\]

\[\frac{100}{3}m\]

Two persons are a metres apart and the height of one is double that of the other. If from the middle point of the line joining their feet, an observer finds the angular elevation of their tops to be complementary, then the height of the shorter post is

\[\frac{a}{4}\]

\[\frac{a}{\sqrt{2}}\]

\[a\sqrt{2}\]

\[\frac{a}{2\sqrt{2}}\]

The angle of elevation of a cloud from a point h metre above a lake is θ. The angle of depression of its reflection in the lake is 45°. The height of the cloud is

*h*tan (45° + θ)*h*cot (45° − θ)*h*tan (45° − θ)*h*cot (45° + θ)

A tower subtends an angle of 30° at a point on the same level as its foot. At a second point *h* metres above the first, the depression of the foot of the tower is 60°. The height of the tower is

\[\frac{h}{2} m\]

\[\sqrt{3h} m\]

\[\frac{h}{3} m\]

\[\frac{h}{\sqrt{3}}m\]

It is found that on walking x meters towards a chimney in a horizontal line through its base, the elevation of its top changes from 30° to 60°. The height of the chimney is

\[3\sqrt{2}x\]

\[2\sqrt{3}x\]

\[\frac{\sqrt{3}}{2}x\]

\[\frac{2}{\sqrt{3}}x\]

The length of the shadow of a tower standing on level ground is found to be 2x metres longer when the sun's elevation is 30°than when it was 45°. The height of the tower in metres is

\[\left( \sqrt{3} + 1 \right) x\]

\[\left( \sqrt{3} - 1 \right) x\]

\[2\sqrt{3}x\]

\[3\sqrt{2}x\]

Two poles are 'a' metres apart and the height of one is double of the other. If from the middle point of the line joining their feet an observer finds the angular elevations of their tops to be complementary, then the height of the smaller is

\[\sqrt{2}a \text{ metres }\]

\[\frac{a}{2\sqrt{2}}\text{ metres }\]

\[\frac{a}{\sqrt{2}} \text{ metres }\]

2a metres

The tops of two poles of height 16 m and 10 m are connected by a wire of length *l*metres. If the wire makes an angle of 30° with the horizontal, then *l* =

26

16

12

10

If a 1.5 m tall girl stands at a distance of 3 m from a lamp-post and casts a shadow of length 4.5 m on the ground, then the height of the lamp-post is

1.5 m

2 m

2.5 m

2.8 m

The length of shadow of a tower on the plane ground is \[\sqrt{3}\] times the height of the tower. The angle of elevation of sun is

45°

30°

60°

90°

The angle of depression of a car, standing on the ground, from the top of a 75 m tower, is 30°. The distance of the car from the base of the tower (in metres) is

\[25\sqrt{3}\]

\[50\sqrt{3}\]

\[75\sqrt{3}\]

150

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 the height of the wall is

\[15\sqrt{3}m\]

\[\frac{15\sqrt{3}}{2}m\]

\[\frac{15}{2}\]

15 m

The angle of depression of a car parked on the road from the top of a 150 m high tower is 30º. The distance of the car from the tower (in metres) is

\[50\sqrt{3}\]

\[150\sqrt{3}\]

\[150\sqrt{2}\]

75

The height of the vertical pole is \[\sqrt{3}\] times the length of its shadow on the ground, then angle of elevation of the sun at that time is

30º

60º

45º

75º

The angle of elevation of the top of a tower at a point on the ground 50 m away from the foot of the tower is 45º. Then the height of the tower (in metres) is

\[50\sqrt{3}\]

50

\[\frac{50}{\sqrt{2}}\]

\[\frac{50}{\sqrt{3}}\]

A ladder makes an angle of 60º with the ground when placed against a wall. If the foot of the ladder is 2 m away from the wall, then the length of the ladder (in metres) is

\[\frac{4}{\sqrt{3}}\]

\[4\sqrt{3}\]

\[2\sqrt{2}\]

\[4\]

## Chapter 12: Trigonometry

## RD Sharma solutions for Class 10 Mathematics chapter 12 - Trigonometry

RD Sharma solutions for Class 10 Mathematics chapter 12 (Trigonometry) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the CBSE Class 10 Mathematics solutions in a manner that help students grasp basic concepts better and faster.

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Concepts covered in Class 10 Mathematics chapter 12 Trigonometry are Trigonometry Ratio of Zero Degree and Negative Angles, Application of Trigonometry, Heights and Distances, Trigonometric Ratios of Complementary Angles, Trigonometric Identities, Trigonometric Ratios in Terms of Coordinates of Point, Angles in Standard Position, Heights and Distances.

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