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SomeApplications of Trigonometry

What is Trigonometry?

Trigonometry is a branch of mathematics that studies triangles and the relationships between their sides and the angles between these sides. 

In this topic we shall make use of Trigonometric Ratios to find the height of a tree, a tower, a water tank, width of a river, distance of ship from lighthouse etc.

Basic Fundamentals

Line of Sight

Line of Sight

Horizontal

We observe generally that children usually look up to see an aero plane when it passes overhead. This line joining their eye to the plane, while looking up is called Line of sight

Line of Sight

Horizontal

Angel of Elevation

The angle which the line of sight makes with a horizontal line

drawn away from their eyes is called the angle of Elevation of

aero plane from them.

Angle of Elevation

• Angle of Elevation: In the picture below, an observer is standing at the top of a building is looking straight ahead (horizontal line). The observer must raise his eyes to see the airplane (slanting line). This is known as the angle of elevation.

• Angle of Depression: The angle below horizontal that an observer must look to see an object that is lower than the observer. Note: The angle of depression is congruent to the angle of elevation (this assumes the object is close enough to the observer so that the horizontals for the observer and the object are effectively parallel).

Angle of Depression

Line of Sight

Horizontal

Angel of Depression

Trigonometric Ratios

Now let us Solve some problem

related toHeight and

Distance

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.

Let AB be the tower and the angle of elevation from point C (on ground) is30°.In ΔABC,

.

Therefore, the height of the tower is

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 °.

Sol:- It can be observed from the figure that AB is the pole. In ΔABC,

Therefore, the height of the pole is 10 m.

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.

Let K be the kite and the string is tied to point P on the ground.In ΔKLP,

.

Hence, the length of the string is

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.

Let AC was the original tree. Due to storm, it was broken into two parts. The broken part

In

,

is making 30° with the ground.

Height of tree =

.

+ BC

Hence, the height of the tree is

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.

Let AQ be the tower and R, S are the points 4m, 9m away from the base of the tower respectively.The angles are complementary. Therefore, if one angle is θ, the other will be 90 − θ.In ΔAQR,

In ΔAQS,

On multiplying equations (i) and (ii), we obtain

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

The angle of elevation of the top of a tower from a point At the foot of the tower is 300 . And after advancing 150mtrs Towards the foot of the tower, the angle of elevation becomes 600 .Find the height of the tower

150

h

d

30 60

mh

h

hh

hh

hh

dofvaluethengSubstituti

hd

From

hdFrom

d

hTan

d

hTan

9.129732.1*75

31502

31503

31503

)1503(3

..........

)150(3

)2(

3)1(

)2(150

360

)1(3

130

Questions based on trigonometry :-• The angle of elevation of the top of a pole measures 48° from a point on

the ground 18 ft. away from its base. Find the height of the flagpole.• Solution Step 1: Let’s first visualize the situationStep 2: Let ‘x’ be the height of the flagpole.STEP 3: From triangle ABC,tan48=x/18Step 4: x = 18 × tan 48° = 18 × 1.11061… = 19.99102…» 20Step 5: So, the flagpole is about 20 ft. high.

45

50

B

A

C

D

A hoarding is fitted above a building. The height of the building is 12 m. When I look at the lights fitted on top of the hoarding, the angle of elevation is 500 and when I look at the top of the building from the same place, the angle is 450. If the height of the flat on each floor is equal to the height of the hoarding, the max floors on the building are? (Tan 500=1.1917)

ANSWER : Let AB denote the height of the building, Let AC denote the height of the hoarding on top of the building Thus, Tan500 = (12 + AC) ÷ 12 1.1917 = 1 + (AC ÷ 12) 1.1917 – 1 = AC ÷ 12 12 ÷ AC = 1 ÷ 0.1917 ~ 5

Made by:-DEEPAK DALAL

Class:-XRoll no.:- 35

The End