This branch of mathematics was invented when ancient mathematicians were trying to learn astronomy (the study of planets, the sun, the moon, and other heavenly bodies) but later on it was found very useful in other fields like geographic studies, navigation, etc. In the year 1852 during the Great Trigonometric survey of British India the highest mountain in the world was discovered from a distance of 160km with help of trigonometric calculations. Today even simpler things like the fall of a tree during a storm can be predicted before using its application. All kinds of building construction applications of trigonometry are important. In other fields like optics and acoustics, it is used to explain sound and light waves, oceanography, medical imaging, etc.
The important question for class 10 maths Chapter 9 Application of Trigonometry can be related to these entities:
- The line of sight
- The angle of elevation
- The angle of depression
- Calculating height and distances using trigonometric calculations
Important Questions Chapter 9 – Some Applications of Trigonometry
Q1. 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 the angle of depression of the base of hill as 30°. Find the distance of the hill from the ship and the height of the hill
Let the man standing on the deck of a ship be at point A and let CE be the hill.
Here BC is the distance of hill from ship and CE be the height of hill.
In rt. ∠ABC, tan 30° = AB/BC
BC = 10√3 m .(i)
BC = 10(1.73) = 17.3 m …[:: √3 = 1.73
AD = BC = 10√3 m …(ii) [From (i)
In rt. ∆ADE, tan 60° = DE/AD
⇒ √3=DE/10√3 … [From (ii)
⇒ DE = 10√3 × √3 = 30 m
∴ CE = CD + DE = 10 + 30 = 40 m
Hence, the distance of the hill from the ship is 17.3 m and the height of the hill is 40 m.
Q2. 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 calculate the height of the wall.
∠BAC = 180° – 90° – 60° = 30°
sin 30° = BC/AC
2BC = 15
BC = 15/2 m
Q3. In the figure, AB is a 6 m high pole and CD is a ladder inclined at an angle of 60° to the horizontal and reaches up to a point D of pole. If AD = 2.54 m. Find the length of the ladder. (Use 3√= 1.73)
BD = AB – AD = 6 – 2.54 = 3.46 m
In rt., ∆DBC, sin 60° = BD/DC
√3/2 = 3.46/DC
√3DC = 3.46 x 2
∴ Length of the ladder, DC = 6.92/√3=6.92/1.73
DC = 4 m
Q4. If a tower 30 m high, casts a shadow 10√3 m long on the ground, then what is the angle of elevation of the sun?
Let required angle be θ.
tan θ = 30/10√3
tan θ = √3
⇒ tan θ = tan 60° ∴ θ = 60°
Q5. 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 = 1.732]
Let BC be the tower
and CD be the flagstaff.
In rt. ∆ABC, tan 45° = BC/AB
1 = BC/120
BC = 120 m …(i)
In rt. ∆ABD,
tan 60° = BD/AB
BD = 120 √3 …(ii)
Height of the flagstaff,
CD = BD – BC
= 120√3 – 120
= 120(√3– 1)
= 120(1.73 – 1)
= 120(0.73) = 87.6 m
Q6. From a point P on the ground the angle of elevation of the top of a tower is 30° and that of the top of a flag staff fixed on the top of the tower, is 60°. If the length of the flag staff is 5m, find the height of the tower.
(i) x/y= tan 30° = 1/√3
y = √3x ..(i)
(ii) (x+5)/y = tan 60°
tan 60° = √3
(x+5)/√3x = √3 …[From (i)
⇒ 3x = x + 5 or x = 2.5
∴ Height of Tower = x = 2.5 m
Q7. When a staircase is lying against a wall, it forms a 60° angle with the horizontal. Calculate the length of the ladder if the foot of the ladder is 2.5 metres from the wall.
Q8. The angle of elevation of the top of a tower from a location 20 metres away is 30 degrees. Determine the tower’s height.
Q9. A flagstaff perches atop a 5m tall structure. The angle of elevation of the top of the flagstaff is 60 degrees from a point on earth, while the angle of elevation of the top of the structure is 45 degrees from the same point. Determine the flagstaff’s height.
Q10. The foot of a tower is reached along a straight highway. A man standing at the top of the tower notices a car approaching the foot of the tower at a consistent pace at a 30° angle of depression. The angle of dip of the automobile is found to be 60 degrees six seconds later. Calculate the time it took the car to go to the foot of the tower from this location.