CBSE Important Questions Class 10 Maths Chapter 9 – Some Applications of Trigonometry

Class 10 Maths Chapter 9 Some Applications of Trigonometry is part of the CBSE Class 10 Maths syllabus. It shows how trigonometry is used in real life to find heights and distances that cannot be measured directly. Students learn the meaning of the line of sight, the angle of elevation and the angle of depression, and how to use the ratios tan, sin and cos to solve such problems.

These CBSE important questions for Class 10 Maths Chapter 9 cover finding the length of a stick leaning against a wall, the height of an electric pole from a tower, a case study on a tower at the corner of a rectangular field, the shadow of a pole at different angles, boats seen from a cliff at equal angles of depression, and assertion-reason questions. Every question has a clear step-by-step solution in simple English with figures, so students can practise on their own, parents can check the working, and teachers can use them for class tests.

Q1. A stick makes an angle of 30° with the ground. If the foot of the stick is 5 m away from the wall, find the length of the stick.

Answer: Let AC be the stick of length h metres and BC = 5 m be the distance of its foot from the wall.
In right △ABC:
cos C = BC / AC
cos 30° = 5 / h
√3/2 = 5 / h
h = 10/√3
Rationalising: h = (10√3)/3 metres
So the length of the stick is (10√3)/3 m, which is about 5.77 m.

Q2. An electric pole and a 15 m high tower stand on level ground. The angle of elevation of the top of the tower from the bottom of the pole is 60°, and from the top of the pole it is 30°. What is the height of the electric pole?
(a) 5 m
(b) 8 m
(c) 10 m
(d) 12 m

Answer: (c) 10 m
Let the distance between the pole and the tower be d, and the height of the pole be h.
From the bottom of the pole: tan 60° = 15/d, so √3 = 15/d, giving d = 15/√3 = 5√3 m.
From the top of the pole: the remaining height of the tower above the pole is (15 − h).
tan 30° = (15 − h)/d
1/√3 = (15 − h)/(5√3)
5√3/√3 = 15 − h
5 = 15 − h
h = 10 m

Q3. A vertical tower of height 4√3 metres stands at one corner of a rectangular field. It subtends angles of 60° and 45° at the two nearest corners of the field.
(i) Find the dimensions of the rectangular field.
(ii) If θ is the angle the tower subtends at the farthest corner, find tanθ.

Answer: Let the length be l and the breadth be b. The height of the tower is 4√3 m.
(i) For the first corner (angle 60°):
tan 60° = height / l
√3 = 4√3 / l
l = 4 m
For the second corner (angle 45°):
tan 45° = height / b
1 = 4√3 / b
b = 4√3 m
So the field is 4 m × 4√3 m.
(ii) The farthest corner is at the end of the diagonal.
Diagonal = √[4² + (4√3)²] = √(16 + 48) = √64 = 8 m
tanθ = height / diagonal = 4√3 / 8 = √3/2

Q4. A monkey is on a tower looking at a tree in the field that is 8 metres tall. The angle of elevation to the top of the tree is 60° and the angle of depression to the bottom of the tree is 30°. Calculate the height of the monkey on the tower.

Answer: Let the monkey be at a height of h metres on the tower, and let the horizontal distance between the tower and the tree be d.
Angle of depression to the bottom of the tree (30°):
tan 30° = h / d
1/√3 = h / d, so d = h√3
Angle of elevation to the top of the tree (60°):
The tree rises (8 − h) metres above the monkey's level.
tan 60° = (8 − h) / d
√3 = (8 − h) / (h√3)
3h = 8 − h
4h = 8
h = 2 m
So the monkey is 2 metres up the tower.

Q5. Read the statements and select the correct option.
Assertion (A): A pole of unknown height casts shadows of two different lengths at different times of the day, corresponding to angles of elevation α and β. If the difference in the shadow lengths is known, the height of the pole can be calculated.
Reason (R): The height of the pole remains constant, and the tangent of the angle of elevation gives the ratio of height to shadow length.
(a) Both A and R are true, and R is the correct explanation of A
(b) Both A and R are true, but R is not the correct explanation of A
(c) A is true but R is false
(d) A is false but R is true

Answer: (a) Both A and R are true, and R is the correct explanation of A
If the height is h, the two shadows are h/tanα and h/tanβ. Their difference is h(1/tanα − 1/tanβ), and since this difference is known, h can be found. The relation tan(angle) = height/shadow is exactly what makes this possible, so R explains A.

Q6. Read the statements and select the correct option.
Assertion (A): From the top of a vertical cliff, the angles of depression of two boats on the sea are observed to be equal. Then the boats are at the same distance from the base of the cliff.
Reason (R): In right-angled triangles with equal angles and equal opposite sides, the adjacent sides must also be equal.
(a) Both A and R are true, and R is the correct explanation of A
(b) Both A and R are true, but R is not the correct explanation of A
(c) A is true but R is false
(d) A is false but R is true

Answer: (a) Both A and R are true, and R is the correct explanation of A
The height of the cliff is the same for both boats. If the angles of depression are equal, then tan(angle) = height/distance gives the same value, so the distances must also be equal. The two right triangles formed are congruent, which is what the Reason states.

Q7. Read the statements and select the correct option.
Assertion (A): A bird is sitting on a tree and an observer on the ground sees it at an angle of elevation θ such that tanθ = 12/5. The distance from the bird to the observer is a multiple of 13.
Reason (R): In a right-angled triangle, (Hypotenuse)² = (Perpendicular)² + (Base)².
(a) Both A and R are true, and R is the correct explanation of A
(b) Both A and R are true, but R is not the correct explanation of A
(c) A is true but R is false
(d) A is false but R is true

Answer: (a) Both A and R are true, and R is the correct explanation of A
Since tanθ = 12/5, the perpendicular and base are in the ratio 12 : 5. Let them be 12k and 5k.
Hypotenuse = √[(12k)² + (5k)²] = √(144k² + 25k²) = √(169k²) = 13k
So the distance from the bird to the observer is 13k, a multiple of 13. The Pythagoras theorem is exactly what gives this result.

Q8. Define the angle of elevation and the angle of depression.

Answer: The line of sight is the straight line drawn from the eye of the observer to the object being viewed.
Angle of elevation: When the object is above the level of the observer's eye, the angle between the line of sight and the horizontal is called the angle of elevation. For example, looking up at the top of a tower.
Angle of depression: When the object is below the level of the observer's eye, the angle between the line of sight and the horizontal is called the angle of depression. For example, looking down at a boat from the top of a cliff.

Q9. 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.

Answer: Let the height of the tower be h metres and the distance from the foot be 30 m.
In the right triangle formed:
tan 30° = h / 30
1/√3 = h / 30
h = 30/√3
Rationalising: h = 30√3/3 = 10√3
So the height of the tower is 10√3 m, which is about 17.32 m.

Q10. A ladder 15 m long just reaches the top of a vertical wall. If the ladder makes an angle of 60° with the wall, find the height of the wall.

Answer: Note carefully that the angle is made with the wall, not with the ground.
So the angle between the ladder and the ground is 90° − 60° = 30°.
Let the height of the wall be h.
sin 30° = h / 15
1/2 = h / 15
h = 7.5 m
So the wall is 7.5 metres high.

Q11. A tree breaks due to a storm and the broken part bends so that the top of the tree touches the ground, making an angle of 30° with it. The distance between the foot of the tree and the point where the top touches the ground is 8 m. Find the height of the tree.

Answer: Let the unbroken part standing upright be h, and the broken part (which now acts as the hypotenuse) be l.
For the standing part:
tan 30° = h / 8
1/√3 = h / 8, so h = 8/√3 m
For the broken part:
cos 30° = 8 / l
√3/2 = 8 / l, so l = 16/√3 m
Total height of the tree = h + l = 8/√3 + 16/√3 = 24/√3
Rationalising: = 8√3 m
So the height of the tree is 8√3 m, which is about 13.86 m.

Q12. 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.

Answer: Let the horizontal distance between the building and the tower be d.
Angle of depression of the foot (45°):
tan 45° = 7 / d
1 = 7 / d, so d = 7 m
Angle of elevation of the top (60°):
Let the part of the tower above the building be x.
tan 60° = x / d
√3 = x / 7, so x = 7√3 m
Total height of the tower = 7 + 7√3 = 7(1 + √3) m, which is about 19.12 m.

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