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Chapter 1:Real Numbers

Chapter 2:Polynomials

Chapter 3:Pair of Linear Equations in Two Variables

Chapter 4:Quadratic Equations

Chapter 5: Arithmetic Progression

Chapter 6: Triangles

Chapter 7:Coordinate Geometry

Chapter 8: Introduction to Trigonometry

Chapter 9:Some Applications of Trigonometry

Chapter 10: Circles

Chapter 11:Constructions

Chapter 12:Area Related to Circles

Chapter 13:Surface Areas and Volumes

Chapter 14:Statistics

Chapter 15:Probability

NCERT Solutions class 10 maths chapter 9

Some Applications of Trigonometry

NCERT Solutions class 10 maths chapter 9 Exercise 9.1

Question 1
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 fig. 9.11)

NCERT Ex-9.1 class 10
Answer

IN the figure

angle made by the rope with the ground level is 30° and Length of the rope is 20 m and we have to determine the height of the pole AB=?
So,.

AC = 20 m and angle θ= 30°

In right ΔABC,

sin 30° = AB/AC

1/2 = AB/20

AB = 20/2

AB = 10

Therefore, the height of the pole is 10 m.

Question 2
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.

NCERT Ex-9.1 class 10
Answer

Let's the original height of tree be AC due to storm tree broke from point .the broken part of tree is BC touch at ground at D making a angle 30°

BC = 8 m

From figure: Total height of the tree is the sum of AB and AC i.e. AB+AC

In right ΔABC,

cos30° = BC/AC

Here, cos 30° = √3/2

√3/2 = 8/AC

AC = 16/√3 …(1)

Also,

tan 30° = AB/BC

1/√3 = AB/8

AB = 8/√3 ….(2)

Therefore, total height of the tree = AB + AC = 16/√3 + 8/√3 = 24/√3 = 8√3 m.

Question 3
A contractor plans to install two slides for the children to play in a park. For the children below the age of 5 years, she prefers to have a slide whose top is at a height of 1.5 m, and is inclined at an angle of 30° to the ground, whereas for elder children, she wants to have a steep slide at a height of 3m, and inclined at an angle of 60° to the ground. What should be the length of the slide in each case?

NCERT Ex-9.1 class 10
Answer

As per contractor’s plan,

Let AC is the slide for the children below the age of 5 yr

So , BC=1.5 ∠CAB=30°

In right ΔABC,

sin 30° = AB/AC

1/2 = 1.5/AC

AC = 3

Let PR is the slide for the children below the age of elder child

So , PQ=3m ∠QPR=60°

Also,

In right ΔPQR,

sin 60° = PQ/PR

⇒ √3/2 = 3/PR

⇒ PR = 2√3

Hence, length of the slide for below 5 = 3 m and

Length of the slide for elders children = 2√3 m

Question 4
The angle of elevation of the top of a tower from a point on the ground, which is 30 m away from the foot of the tower, is 30°. Find the height of the tower.

NCERT Ex-9.1 class 10
Answer

Let AB be the height of the tower and C is the point elevation which is 30 m away from the foot of the tower.

here ∠ACB=30°

In right ABC

tan 30° = AB/BC

1/√3 = AB/30

⇒ AB = 10√3

Thus, the height of the tower is 10√3 m.

Question 5
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°. Find the length of the string, assuming that there is no slack in the string.

NCERT Ex-9.1 class 10
Answer

Let BC is Height of the kite from the ground, BC = 60 m

AC is length of the string from the ground which make an angle ∠ACB=60°

From the above figure,

sin 60° = BC/AC

⇒ √3/2 = 60/AC

⇒ AC = 40√3 m

Thus, the length of the string from the ground is 40√3 m.

Question 6
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.

NCERT Ex-9.1 class 10
Answer

Let the boy initially stand at point Y with inclination 30° and then he approaches the building to the point X with inclination 60°.

From figure,

XY = CD.

Height of the building = AZ = 30 m.

AB = AZ – BZ = 30 – 1.5 = 28.5

Measure of AB is 28.5 m

In right ΔABD,

tan 30° = AB/BD

1/√3 = 28.5/BD

BD = 28.5√3 m

Again,

In right ΔABC,

tan 60° = AB/BC

√3 = 28.5/BC

BC = 28.5/√3 = 28.5√3/3

Therefore, the length of BC is 28.5√3/3 m.

XY = CD = BD – BC = (28.5√3-28.5√3/3) = 28.5√3(1-1/3) = 28.5√3 × 2/3 = 57/√3 m.

Thus, the distance boy walked towards the building is 57/√3 m.

Question 7
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.

NCERT Ex-9.1 class 10
Answer

Let BC be the 20 m high building.

D is the point on the ground from where the elevation is taken.

Height of transmission tower = AB = AC – BC

From figure, In right ΔBCD,

tan 45° = BC/CD

1 = 20/CD

CD = 20

Again,

In right ΔACD,

tan 60° = AC/CD

√3 = AC/20

AC = 20√3

Now, AB = AC – BC = (20√3-20) = 20(√3-1)

Height of transmission tower = 20(√3 – 1) m.

Question 8
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.

NCERT Ex-9.1 class 10
Answer

Let AB be the height of statue.

D is the point on the ground from where the elevation is taken.

To Find: Height of pedestal = BC = AC-AB

From figure,

In right triangle BCD,

tan 45° = BC/CD

1 = BC/CD

BC = CD …..(1)

Again,

In right ΔACD,

tan 60° = AC/AD

√3 = ( AB+BC)/CD

√3CD = 1.6 + BC

√3BC = 1.6 + BC (using equation (1)

√3BC – BC = 1.6

BC(√3-1) = 1.6

BC = 1.6/(√3-1) m

BC = 0.8(√3+1)

Thus, the height of the pedestal is 0.8(√3+1) m.

Question 9
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.

NCERT Ex-9.1 class 10
Answer

Let CD be the height of the tower. AB be the height of the building. BC be the distance between the foot of the building and the tower. Elevation is 30 degree and 60 degree from the tower and the building respectively.

In right ΔBCD,

tan 60° = CD/BC

√3 = 50/BC

BC = 50/√3 …(1)

Again,

In right ΔABC,

tan 30° = AB/BC

1/√3 = AB/BC

Use result obtained in equation (1)

AB = 50/3

Thus, the height of the building is 50/3 m.

Question 10
Two poles of equal heights are standing opposite 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.

NCERT Ex-9.1 class 10
Answer

Let AB and CD be the poles of equal height.

O is the point between them from where the height of elevation taken. BD is the distance between the poles.

As per above figure, AB = CD,

OB + OD = 80 m

Now,

In right ΔCDO,

tan 30° = CD/OD

1/√3 = CD/OD

CD = OD/√3 … (1)

Again,

In right ΔABO,

tan 60° = AB/OB

√3 = AB/(80-OD)

AB = √3(80-OD)

AB = CD (Given)

√3(80-OD) = OD/√3 (Using equation (1))

3(80-OD) = OD

240 – 3 OD = OD

4 OD = 240

OD = 60

Putting the value of OD in equation (1)

CD = OD/√3

CD = 60/√3

CD = 20√3 m

Also,

OB + OD = 80 m

OB = (80-60) m = 20 m

Thus, the height of the poles are 20√3 m and distance from the point of elevation are 20 m and

60 m respectively.

Question 11
A TV tower stands vertically on a bank of a canal. From a point on the other bank directly opposite the tower, the angle of elevation of the top of the tower is 60°. From another point 20 m away from this point on the line joing this point to the foot of the tower, the angle of elevation of the top of the tower is 30° (see Fig. 9.12). Find the height of the tower and the width of the canal.

NCERT Ex-9.1 class 10

Answer
Given, AB is the height of the tower.

DC = 20 m (given)

As per given diagram, In right ΔABD,

tan 30° = AB/BD

1/√3 = AB/(20+BC)

AB = (20+BC)/√3 … (i)

Again,

In right ΔABC,

tan 60° = AB/BC

√3 = AB/BC

AB = √3 BC … (ii)

From equation (i) and (ii)

√3 BC = (20+BC)/√3

3 BC = 20 + BC

2 BC = 20

BC = 10

Putting the value of BC in equation (ii)

AB = 10√3

This implies, the height of the tower is 10√3 m and the width of the canal is 10 m.

Question 12
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.

NCERT Ex-9.1 class 10
Answer

Let AB be the building of height 7 m and EC be the height of the tower.

A is the point from where elevation of tower is 60° and the angle of depression of its foot is 45°.

EC = DE + CD

Also, CD = AB = 7 m. and BC = AD

To Find: EC = Height of the tower

Design a figure based on given instructions:

In right ΔABC,

tan 45° = AB/BC

1= 7/BC

BC = 7

Since BC = AD

So AD = 7

Again, from right triangle ADE,

tan 60° = DE/AD

√3 = DE/7

⇒ DE = 7√3 m

Now: EC = DE + CD

= (7√3 + 7) = 7(√3+1)

Therefore, Height of the tower is 7(√3+1) m. Answer!

Question 13
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.

NCERT Ex-9.1 class 10
Answer

Let AB be the lighthouse of height 75 m. Let C and D be the positions of the ships.

30° and 45° are the angles of depression from the lighthouse.

Draw a figure based on given instructions:

To Find: CD = distance between two ships

Step 1: From right triangle ABC,

tan 45° = AB/BC

1= 75/BC

BC = 75 m

Step 2: Form right triangle ABD,

tan 30° = AB/BD

1/√3 = 75/BD

BD = 75√3

Step 3: To find measure of CD, use results obtained in step 1 and step 2.

CD = BD – BC = (75√3 – 75) = 75(√3-1)

The distance between the two ships is 75(√3-1) m. Answer!

Question 14
A 1.2 m tall girl spots a balloon moving with the wind in a horizontal line at a height of 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 some time, the angle of elevation reduces to 30° (see Fig. 9.13). Find the distance travelled by the balloon during the interval.

NCERT Ex-9.1 class 10

Answer
NCERT Ex-9.1 class 10

Let the initial position of the balloon be A and final position be B.

Height of balloon above the girl height = 88.2 m – 1.2 m = 87 m.

To Find: Distance travelled by the balloon = DE = CE – CD

Let us redesign the given figure as per our convenient

tan 30° = BE/CE

1/√3= 87/CE

CE = 87√3

Step 2:

In right ΔADC,

tan 60° = AD/CD

√3= 87/CD

CD = 87/√3 = 29√3

Step 3:

DE = CE – CD = (87√3 – 29√3) = 29√3(3 – 1) = 58√3

Distance travelled by the balloon = 58√3 m.

Question 15
A straight highway leads to the foot of a tower. A man standing at the top of the tower observes a car at an angle of depression of 30°, which is approaching the foot of the tower with a uniform speed. Six seconds later, the angle of depression of the car is found to be 60°. Find the time taken by the car to reach the foot of the tower from this point.

NCERT Ex-9.1 class 10
Answer

Let AB be the tower.

D is the initial and C is the final position of the car respectively.

Since man is standing at the top of the tower so, Angles of depression are measured from A.

BC is the distance from the foot of the tower to the car.

Step 1: In right ΔABC,

tan 60° = AB/BC

√3 = AB/BC

BC = AB/√3

AB = √3 BC

Step 2:

In right ΔABD,

tan 30° = AB/BD

1/√3 = AB/BD

AB = BD/√3

Step 3: Form step 1 and Step 2, we have

√3 BC = BD/√3 (Since LHS are same, so RHS are also same)

3 BC = BD

3 BC = BC + CD

2BC = CD

or BC = CD/2

Here, distance of BC is half of CD. Thus, the time taken is also half.

Time taken by car to travel distance CD = 6 sec. Time taken by car to travel BC = 6/2 = 3 sec.

Question 16
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.>

NCERT Ex-9.1 class 10
Answer

Let AB be the tower. C and D be the two points with distance 4 m and 9 m from the base respectively. As per question,

In right ΔABC,

tan x = AB/BC

tan x = AB/4

AB = 4 tan x … (i)

Again, from right ΔABD,

tan (90°-x) = AB/BD

cot x = AB/9

AB = 9 cot x … (ii)

Multiplying equation (i) and (ii)

AB2 = 9 cot x × 4 tan x

⇒ AB2 = 36 (because cot x = 1/tan x

⇒ AB = ± 6

Since height cannot be negative. Therefore, the height of the tower is 6 m.

Hence Proved.