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Tamilnadu Samacheer Kalvi 11th Maths Solutions Chapter 3 Trigonometry Ex 3.2

Question 1.
Express each of the following angles in radian measure:
(i) 30°
(ii) 135°
(iii) -205°
(iv) 150°
(v) 330°
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 3 Trigonometry Ex 3.2 1

Question 2.
Find the degree measure corresponding to the following radian measures
Samacheer Kalvi 11th Maths Solutions Chapter 3 Trigonometry Ex 3.2 2
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 3 Trigonometry Ex 3.2 3

Samacheer Kalvi 11th Maths Solutions Chapter 3 Trigonometry Ex 3.2

Question 3.
What must be the radius of a circular running path, around which an athlete must run 5 times in order to describe 1 km?
Solution:
Distance travelled in 5 rounds = 1 km = 1000 m
Distance travelled in 1 round = \(\frac{1000}{5}\) = 200 m
Let the radius of the circular path be r metre
So 2πr = 200
Samacheer Kalvi 11th Maths Solutions Chapter 3 Trigonometry Ex 3.2 4

Question 4.
In a circle of diameter 40 cm, a chord is of length 20 cm. Find the length of the minor arc of the chord.
Solution:
O = centre of the circle
PQ = diameter = 40 cm
∴ OQ = 20 cm
radius = 20 cm
⇒ OA = OB = 20 cm
chord AB = 20 cm
OC ⊥ r AB
∴ AC = CB = 10 cm
Now from the right angled triangle OCB
Samacheer Kalvi 11th Maths Solutions Chapter 3 Trigonometry Ex 3.2 20
Samacheer Kalvi 11th Maths Solutions Chapter 3 Trigonometry Ex 3.2 21

Samacheer Kalvi 11th Maths Solutions Chapter 3 Trigonometry Ex 3.2

Question 5.
Find the degree measure of the angle subtended at the centre of circle of radius 100 cm by an arc of length 22 cm.
Solution:
r = 100 cm; arc length = 22 cm
Samacheer Kalvi 11th Maths Solutions Chapter 3 Trigonometry Ex 3.2 22
Question 6.
What is the length of the arc intercepted by a central angle of measure 41° in a circle of radius 10 ft?
Solution:
θ = 41°, r = 10 ft
Samacheer Kalvi 11th Maths Solutions Chapter 3 Trigonometry Ex 3.2 23

Question 7.
If in two circles, arcs of the same length subtend angles 60° and 75° at the centre, find the ratio of their radii.
Solution:
Let the two radii be r1 and r2
The central angles arc 60° and 75°
The arc lengths be s1 and s2
Samacheer Kalvi 11th Maths Solutions Chapter 3 Trigonometry Ex 3.2 24
So their radii are in the ratio 5 : 4.

Question 8.
The perimeter of a certain sector of a circle is equal to the length of the arc of a semicircle having the same radius. Express the angle of the sector in degrees, minutes and seconds.
Solution:
Let r be the radius and so perimeter of a sector = l + 2r
Length of arc of the semicircle = πr
we are given l + 2r = πr
(i.e) l = πr – 2r
Samacheer Kalvi 11th Maths Solutions Chapter 3 Trigonometry Ex 3.2 25
Samacheer Kalvi 11th Maths Solutions Chapter 3 Trigonometry Ex 3.2 26

Question 9.
An airplane propeller rotates 1000 times per minute. Find the number of degrees that a point on the edge of the propeller will rotate in 1 second.
Solution:
Number of rotations in 1 min = 1000
Samacheer Kalvi 11th Maths Solutions Chapter 3 Trigonometry Ex 3.2 28
The angle rotated in 1 rotation = 360°
Samacheer Kalvi 11th Maths Solutions Chapter 3 Trigonometry Ex 3.2 29

Samacheer Kalvi 11th Maths Solutions Chapter 3 Trigonometry Ex 3.2

Question 10.
A train is moving on a circular track of 1500 m radius at the rate of 66 km / hr. What angle will it turn in 20 seconds?
Solution:
Speed of the train = 66 km/hr
Samacheer Kalvi 11th Maths Solutions Chapter 3 Trigonometry Ex 3.2 30

Question 11.
A circular metallic plate of radius 8 cm and thickness 6 mm is melted and molded into a pie (a sector of the circle with thickness) of radius 16 cm and thickness 4 mm. Find the angle of the sector.
Solution:
Area of the circular plate melted
Samacheer Kalvi 11th Maths Solutions Chapter 3 Trigonometry Ex 3.2 31
Samacheer Kalvi 11th Maths Solutions Chapter 3 Trigonometry Ex 3.2 32

Samacheer Kalvi 11th Maths Solutions Chapter 3 Trigonometry Ex 3.2 Additional Questions

Question 1.
An athlete runs 4 times around a circular running track to describe 1760 m. What is the (radius of the tract) degrees subtended at the centre of the circle, after he has run a distance of 308 m?
Solution:
[Hint: Distance travelled in 4 rounds = 1760 m]
Samacheer Kalvi 11th Maths Solutions Chapter 3 Trigonometry Ex 3.2 33

Samacheer Kalvi 11th Maths Solutions Chapter 3 Trigonometry Ex 3.2

Question 2.
Find the angle through which a pendulum swings if its length is 75 cm and the tip describes an arc of length
(i) 10 cm
(ii) 15 cm
(iii) 19 cm
Solution:
Here, r = 75 cm
Samacheer Kalvi 11th Maths Solutions Chapter 3 Trigonometry Ex 3.2 34

Question 3.
The angles of a quadrilateral are in A.P and the greatest angle is 120°. Express the other angles in radians.
Solution:
Let the angles of the quadrilateral be a°, (a + d)°, (a + 2d)° and (a + 3d)°.
Since sum of all angles of a quadrilateral is 360°, we have .
a° + (a + d)° + (a + 2 d)° + (a + 3 d)° = 360° ⇒ 4a + 6d = 360°
⇒ 2a + 3d = 180° …(1)
Now, greatest angle is 120°
So, a + 3d = 120° …(2)
Solving (1) and (2), we have
a = 60°, d = 20°
Hence, the angles are 60°, 80°, 100°, 120°.
Samacheer Kalvi 11th Maths Solutions Chapter 3 Trigonometry Ex 3.2 60
Samacheer Kalvi 11th Maths Solutions Chapter 3 Trigonometry Ex 3.2 61

Question 4.
A railroad curve is to be laid out on a circle. What radius should be used if the track is to change direction by 25° in a distance of 40 metres?
Solution:
Let the radius of the circle on which the railroad curve is to be laid down be x metres and the angle subtended by it at the centre is
Samacheer Kalvi 11th Maths Solutions Chapter 3 Trigonometry Ex 3.2 62

Question 5.
A horse is tied to a post by a rope. If the horse moves along a circular path always keeping the rope tight and describes 88 metres when it has traced out 72° at the centre, find the length of the rope.
Solution:
Let O be the post.
Let A, B be the two positions of the horse.
Here l, the length of the arc AB = 88 metres
Angle subtended = 72°
Samacheer Kalvi 11th Maths Solutions Chapter 3 Trigonometry Ex 3.2 63
Hence, the length of the rope = 70 metres

Samacheer Kalvi 11th Maths Solutions Chapter 3 Trigonometry Ex 3.2

Question 6.
A circular wire of radius 3 cm is cut and bent so as to lie along the circumference of a sector whose radius is 48 cm. Find in degrees the angle which is subtended at the centre of the sector.
Solution:
Length of arc = Circumference of wire of radius = 3 cm
l = 2πr = 2π × 3 = 6π cm
The radius of the sector (r) = 48 cm
Samacheer Kalvi 11th Maths Solutions Chapter 3 Trigonometry Ex 3.2 64

Question 7.
Samacheer Kalvi 11th Maths Solutions Chapter 3 Trigonometry Ex 3.2 65
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 3 Trigonometry Ex 3.2 66

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