© Ck Coaching, Hakim Para, Siliguri-734001, Ph-8116896002

PRACTICE SHEET-1 {2022-23}

MATHEMATICS-GRADE-10 [CBSE]

SECTION: TRIGONOMETRY – 12 Marks

NAME: _______________________
Contact: _______________________
 © Ck Coaching, Hakim Para, Siliguri-734001, Ph-8116896002
1-Marks MCQ
𝑎
Q.1 Given that sin 𝜃 = 𝑏, then cos 𝜃 is Q.9 The value of (sin 45° + cos 45° )is
1
(a)
𝑏 𝑏
(b) 𝑎 (a) (b) √2
√2
√𝑏 2 −𝑎2
√3
√𝑏 2 −𝑎2 𝑎 (c) (d) 1
(c) (d) √𝑏2 2
𝑏 −𝑎2
4
1 1 Q.10 If cos 𝜃 = , then find the value of tan 𝜃
Q.2 If 𝑠𝑖𝑛𝐴 = 2 𝑎𝑛𝑑 cos 𝐵 = 2, then the value 5

of (A+B) is 3 3
(a) 5 (b) 4
(a) 0° (b) 30° 4 5
(c) (d) 3
3
(c) 60° (d) 90°
Q.11 If sin 𝐴 + 𝑠𝑖𝑛2 𝐴 = 1, then the value of the
Q.3 If 𝑥 𝑡𝑎𝑛60° cos 60° = sin 60° cot 60°, then
expression (𝑐𝑜𝑠 2 𝐴 + 𝑐𝑜𝑠 4 𝐴) 𝑖𝑠
the value of x
1
(a) cos 30° (b) tan 30° (a) 1 (b) 2

(c) sin 30° (d) 𝑐𝑜𝑡30° (c) 2 (d) 3

Q.4 Given that, 𝑠𝑖𝑛∅ + 𝑐𝑜𝑠∅ = √2, then Q.12 A pole 6 m high casts a shadow 2√3 𝑚,
tan ∅ + cot ∅ = long on the ground, the sun’s elevation is

(a) 1 (b) 2 (a) 30° (b) 45°

(c) 3 (d) 4 (c) 60° (d) 90°

5𝑠𝑖𝑛𝛼−2𝑐𝑜𝑠𝛼 Q.13 If sin 𝜃 + cos 𝜃 = √2 cos 𝜃, ( 𝜃 ≠ 90°)then
Q.5 If 5 𝑡𝑎𝑛𝛼 = 4 , then =
5𝑠𝑖𝑛𝛼+2𝑐𝑜𝑠𝛼 the value of tan 𝜃 is
1 2
(a) 3 (b) 5 (a) √2 − 1 (b) √2 + 1
11
(c) 36 (d) 6 (c) √2 (d) −√2
√3
Q.6 The angle of depression of a car parked on Q.14 Given that sin 𝛼 = 𝑎𝑛𝑑 cos 𝛽 = 0, the
2
the road from the top of 150m high tower is value of 𝛽 − 𝛼 is
30°. The distance of the car from the tower
(in m) is: (a) 0° (b) 90°

(a) 50√3 (b) 150√3 (c) 60° (d) 30°

(c) 150√2 (d) 75 Q.15 If sin 𝜃 = cos 𝜃, then the value of

Q.7 In the adjoining figure, the length of BC is 𝑡𝑎𝑛2 𝜃 + 𝑐𝑜𝑡 2 𝜃 is

(a) 2 (b) 4
10
(c) 1 (d) 3

Q.16 If in ∆ 𝐴𝐵𝐶 , < 𝐵 = 90°, then the value of

tan(𝐴 + 𝐶).
1
(a) (b) √3
√3

(a) 2√3 𝑐𝑚 (b) 3√3 𝑐𝑚 (c) 1 (d) Not defined

(c) 4√3 𝑐𝑚 (d) 3cm Q.17 If 2 sin 𝐴 = 1, then the value of tan 𝐴 𝑖𝑠
1
Q.8 The height of a tree, if it casts a shadow 15 (a) (b) √3
√3
m long on the level of ground, when the
angle of elevation of the sun is 45°, is (c) ½ (d) 1

(a) 10 m (b) 14 m Q.18 The ratio of the length of a rod and its
shadow is 1 ∶ √3 , the angle of elevation of
(c) 8 m (d) 15 m
the sum is
(a) 30° (b) 45°
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(c) 60° (d) 90° SAQ (2/3 marks)
Q.19 The top of two poles of height 16 m and 10 𝑐𝑜𝑡 2 ∅
Q.1 Prove that: 1 + 1+𝑐𝑜𝑠𝑒𝑐 ∅ = 𝑐𝑜𝑠𝑒𝑐 ∅
m are connected by a wire of length “l”
metres, if the wire makes an angle of 30° 2 𝑐𝑜𝑠 3 𝜃−𝑐𝑜𝑠𝜃
with the horizontal, then 𝑙 = ? Q.2 Prove that: = cot 𝜃
sin 𝜃−2𝑠𝑖𝑛3 𝜃

(a) 26 m (b) 16 m 2𝑐𝑜𝑠 2 𝐴−1

Q.3 Prove that: 𝑐𝑜𝑡𝐴 − tan 𝐴 =
𝑠𝑖𝑛𝐴 .cos 𝐴
(c) 12 m (d) 10 m
sin 𝐴−2 𝑠𝑖𝑛3 𝐴
Q.20 The length of shadow of a tower on the Q.4 Prove that: = tan 𝐴
2𝑐𝑜𝑠 3 𝐴−𝑐𝑜𝑠𝐴
plane ground is √3 times the height of the 𝑐𝑜𝑒𝑠𝑐𝐴 𝑐𝑜𝑠𝑒𝑐𝐴
tower. The angle of elevation of sun is Q.5 Prove that: 𝑐𝑜𝑠𝑒𝑐𝐴−1 + 𝑐𝑜𝑠𝑒𝑐𝐴+1 = 2 𝑠𝑒𝑐 2 𝐴

(a) 45° (b) 30° Q.6 Evaluate:

3𝑡𝑎𝑛2 30°+𝑡𝑎𝑛2 60°+𝑐𝑜𝑠𝑒𝑠𝑐30°−𝑡𝑎𝑛45°
(c) 60° (d) 90° 𝑐𝑜𝑡 2 45°
1+tan 𝜃 1
Q.21 If sec 𝜃 = √2, the value of Q.7 If sin(𝐴 + 𝐵) = 1 𝑎𝑛𝑑 sin(𝐴 − 𝐵) =
sin 𝜃 2,

(a) 2√2 (b) √2 0 ≤ 𝐴 + 𝐵 < 90°, then find A and B.

√3 √3
(c) 3√2 (d) 2 Q.8 If cos(𝐴 − 𝐵) = 𝑎𝑛𝑑 sin(𝐴 + 𝐵) = ,
2 2
Q.22 Evaluate: find the value of A and B.

𝑠𝑖𝑛2 60° + 2𝑡𝑎𝑛45° − 𝑐𝑜𝑠 2 30° = _______ Q.9 Find the value of cos 2𝜃 , 𝑖𝑓 2𝑠𝑖𝑛2𝜃 = √3.

Q.23 If √2 𝑠𝑖𝑛𝜃 − 𝑐𝑜𝑠𝜃 = 0 , 0° < 𝜃 < 90°, the (𝑠𝑖𝑛4 𝜃+𝑐𝑜𝑠4 𝜃)

Q.10 Prove that: =1
value of 𝜃 is _________________ 1−2𝑠𝑖𝑛2 𝜃𝑐𝑜𝑠2 𝜃

Q.24 If 4𝑡𝑎𝑛𝜃 = 3, then the value of Q.11

1−𝑐𝑜𝑠𝐴
Prove that: √1+𝑐𝑜𝑠𝐴 = 𝑐𝑜𝑠𝑒𝑐𝐴 − 𝑐𝑜𝑡𝐴
4𝑠𝑖𝑛𝜃−𝑐𝑜𝑠𝜃+1
(4 sin 𝜃+cos 𝜃−1)= ____________
Q.12**The rod of TV disc antenna is fixed at right
1
Q.25 The value of angles to wall AB and a rod CD is
tan 𝜃+cot 𝜃 supporting the disc as shown in Figure. If
(a) cos 𝜃 . sin 𝜃 (b) sec 𝜃. 𝑠𝑖𝑛𝜃 AC = 1 5. m long and CD = 3 m, find

(c) 𝑡𝑎𝑛𝜃. 𝑐𝑜𝑡𝜃 (d)𝑠𝑒𝑐𝜃. 𝑐𝑜𝑠𝑒𝑐𝜃 (i) tan θ (ii) sec θ + cosec θ.

Q.26 The value of 9𝑠𝑒𝑐 2 𝐴 − 9𝑡𝑎𝑛2 𝐴 𝑖𝑠

(a) 1 (b) 9
(c) 8 (d) 0
Q.27 If 𝑥 = 2 𝑠𝑖𝑛2 𝜃 𝑎𝑛𝑑 𝑦 = 2𝑐𝑜𝑠 2 𝜃 + 1,then
find the value of 𝑥 + 𝑦 𝑖𝑠
(a) 2 (b) 0
(c) 3 (d) 1
1
Q.28 The value of (𝑠𝑖𝑛2 𝜃 + 1+𝑡𝑎𝑛2 𝜃) is Q.13 If sin 𝜃 + cos 𝜃 = √3, prove that

(a) 2 (b) 1 tan 𝜃 + 𝑐𝑜𝑡𝜃 = 1.

(c) 0 (d) 3 Q.14 An observer, 1.7 m tall, is 20√3 m away

from a tower. The angle of elevation from
Q.29 If 𝑘 + 1 = 𝑠𝑒𝑐 2 𝜃(1 + 𝑠𝑖𝑛𝜃)(1 − 𝑠𝑖𝑛𝜃),
the eye of observer to the top of tower is 30º.
then the value of ‘k’ is
Find the height of tower.
(a) 1 (b) -1
Q.15 An observer 1.5 m tall is 28.5 m away from
(c) 0 (d) 2 a tower 30 m high. Find the angle of
elevation of the top of the tower from his
Q.30 The value of (𝑠𝑒𝑐𝐴 + 𝑡𝑎𝑛𝐴) (1 − 𝑠𝑖𝑛𝐴)is eye.
(a) sec 𝐴 (b) 𝑠𝑖𝑛𝐴
(c) 𝑐𝑜𝑠𝑒𝑐𝐴 (d) 𝑐𝑜𝑠𝐴
 © Ck Coaching, Hakim Para, Siliguri-734001, Ph-8116896002
Q.16 From a point P on the ground the angle of Q.25 The angle of elevation of a cloud from a
elevation of the top of a 10 m tall building is point 120 m above a lake is 30 ° and the
30°. A flag is hoisted at the top the of the angle of depression of its reflection in the
building and the angle of elevation of the lake is 60 °. Find the height of the cloud.
length of the flagstaff from P is 45°. Find the
Q.26 From the top of a tower, 100 m high, a man
length of the flagstaff and distance of
observes two cars on the opposite sides of
building from point P. [Take√3 = 1.732]
the tower and in same straight line with its
Q.17 If the shadow of a tower is 30 m long, when base, with angles of depression 30 ° and 45°.
the Sun’s elevation is 30°. What is the length Find the distance between the cars.
of the shadow, when Sun’s elevation is 60°?
[Take√3 = 1.732]
Q.18 The angle of elevation of the top of a
Q.27 Prove that:
building from the foot of the tower is 30°
and the angle of elevation of the top of the 2(𝑠𝑖𝑛6 𝜃 + 𝑐𝑜𝑠 6 𝜃) − 3 (𝑠𝑖𝑛4 𝜃 + 𝑐𝑜𝑠 4𝜃 ) + 1 = 0
tower from the foot of the building is 45°. If
the tower is 30 m high, find the height of the Q.28 In the given fig. ∆𝑃𝑄𝑅, right-angled at Q,
building. 𝑄𝑅 = 9𝑐𝑚 𝑎𝑛𝑑 𝑃𝑅 − 𝑃𝑄 1𝑐𝑚.Determine
the value of 𝑠𝑖𝑛𝑅 + 𝑐𝑜𝑠𝑅.
Q.19 An aeroplane, when flying at a height of
4000 m from the ground passes vertically
above another aeroplane at an instant when
the angles of elevation of the two planes
from the same point on the ground are 60º
and 45º respectively. Find the vertical
distance between the aeroplanes at that
instant. [Take√3 = 1.732]
Q.20 The horizontal distance between two towers Q.29 In ∆ 𝐴𝐵𝐶 ; < 𝐵 = 90°, 𝐵𝐶 = 5𝑐𝑚,
1+𝑠𝑖𝑛𝐶
is 60 m. The angle of elevation of the top of 𝐴𝐶 − 𝐴𝐵 = 1 , then evaluate:
the taller tower as seen from the top of the 1+𝑐𝑜𝑠𝐶
shorter one is 30º. If the height of the taller Q.30 If 𝑥𝑠𝑖𝑛3 𝜃 + 𝑦𝑐𝑜𝑠 3 𝜃 = 𝑠𝑖𝑛𝜃. 𝑐𝑜𝑠𝜃 and also
tower is 150 m, then find the height of the
shorter tower. 𝑥𝑠𝑖𝑛 𝜃 = 𝑦 cos 𝜃, then prove that

Q.21 From the top of a 7 m high building the 𝑥 2 + 𝑦 2 = 1.

angle of elevation of the top of a tower is Q.31 Prove that:
60 º and the angle of depression of its foot is
sin 𝜃 sin 𝜃
45 º. Determine the height of the tower. =2+
cot 𝜃+𝑐𝑜𝑠𝑒𝑐 𝜃 cot 𝜃−𝑐𝑜𝑠𝑒𝑐 𝜃
Q.22 A man in a boat rowing away from a light
Q.32 If 𝑐𝑜𝑠𝑒𝑐 𝜃 + cot 𝜃 = 𝑝, then prove that
house 100 m high takes 2 minutes to change
the angle of elevation of the top of the light 𝑝2 −1
cos 𝜃 = .
house from 60 º to 30 º. Find the speed of 𝑝2 +1

the boat in metres per minute.

[Take√3 = 1.732] NOTE: FOR PROOF BASED

QUESTION PRACTICE NCERT
Q.23 Amit, standing on a horizontal plane, find a
ASWELL.
bird flying at a distance of 200 m from him
at an elevation of 30 º. Deepak standing on
the roof of a 50 m high building, find the
angle of elevation of the same bird to be
45 º. Amit and Deepak are on opposite sides
of the bird. Find the distance of the bird
from Deepak. {** only for standard}
Q.24 A man on the top of a vertical tower
observes a car moving at a uniform speed
towards him. If it takes 12 min. for the angle
of depression to change from 30 º to 45 º,
how soon after this, the car will reach the
tower? {** only for standard}
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CASE BASED QUESTIONS

Q.1 The given picture below shows three kites Observe the picture. From a point A h m
flying together. above from water level, the angle of
elevation of top of Chhatri (point B) is
45°and angle of depression of its reflection
in water (point C) is 60° . If the height of
Chhatri above water level is (approximately)
10 m, then
(a) draw a well-labelled figure based on the
The angle of elevation of two kites (Points A above information;
and B) from hands of a man (point C) are (b) find the height (h) of the point A above
found to be 30° 𝑎𝑛𝑑 60° respectively. water level.
Taking AD = 50m and BE= 60 m, find
Q.4 A lighthouse is a tall tower with light near
(a) The lengths of the strings used (take the top. These are often built on islands,
them straight) for kites A and B as coasts or on cliffs. Lighthouses on water
shown in the fig. surface act as a navigational aid to the
(b) The distance ‘d’ between these two kites. mariners and send warning to boats and
ships for dangers. Initially wood, coal would
Q.2 Qutub Minar, located in south Delhi, was be used as illuminators. Gradually it was
built in the year 1193. It is 72 m high tower. replaced by candles, lanterns, electric lights.
Anit and Charu visited this monument for Nowadays they are run by machines and
their school project. The used the concept of remote monitoring. Prongs Reef lighthouse
trigonometry to calculate their distance from of Mumbai was constructed in 1874-75. It is
the tower. Point C and D represent their approximately 40 meters high and its beam
positions on the ground in line with the base can be seen at a distance of 30 kilometres. A
of tower, the angles of the top of the tower ship and a boat are coming towards the
(point A) are 60° 𝑎𝑛𝑑 45° from the points C lighthouse from opposite directions. Angles
and D, respectively. of depression of flash light from the
lighthouse to the boat and the ship are
30°and 60° respectively.

i) Which of the two, boat or the ship is

(a) Based on the above information, draw a nearer to the light house. Find its
well-labelled diagram. distance from the lighthouse?
(b) Find the distance CD, BC and BD. ii) Find the time taken by the boat to
(use √3 = 1.73) reach the light house if it is moving
at the rate of 20 km per hour.
Q.3 Gadisar Lake is located in the Jaisalmer
district of Rajasthan. It was built by the King ********************************************
of Jaisalmer and rebuilt by Gadsi Singh in
14th century. The lake has many Chhatris.
One of them is shown below: