Tamilnadu State Board New Syllabus Samacheer Kalvi 11th Maths Guide Pdf Chapter 3 Trigonometry Ex 3.2 Text Book Back Questions and Answers, Notes.

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

Question 1.

Express each of the following in radian measure.

(i) 30°

(ii) 135°

(iii) -205°

(iv) 150°

(v) 330°

Answer:

(i) 30°

(ii) 135°

(iii) – 205°

(iv) 150°

(v) 330°

Question 2.

Find the degree measure corresponding to the following radian measures.

(i) \(\frac{\pi}{3}\)

(ii) \(\frac{\pi}{9}\)

(iii) \(\frac{2 \pi}{5}\)

(iv) \(\frac{7 \pi}{3}\)

(v) \(\frac{10 \pi}{9}\)

Answer:

(i) \(\frac{\pi}{3}\) radians

(ii) \(\frac{\pi}{9}\) radians

(iii) \(\frac{2 \pi}{5}\) radians

(iv) \(\frac{7 \pi}{3}\) radians

(v) \(\frac{10 \pi}{9}\) radians

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?

Answer:

Let the radius of the circular path be = r m.

Length of the circular path s = 1 k. m

s = 1000 m.

Athlete runs 5 times around the path to cover 1 k. m distance

∴ θ = 360° × 5

θ = 360° × 5 × \(\frac{\pi}{180}\) radians

θ = 10 π radians

s = r θ

1000 = r 10 π

r = \(\frac{1000}{10 \pi}\)

r = \(\frac{1000 \times 7}{10 \times 22}=\frac{350}{11}\)

r = 31 .818 meters

Radius of the circular path = 31 .82 meters

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.

Answer:

Given Diameter AB = 40 cm

∴ Radius r = 20 cm

Chord CD = 20 cm

O – Centre of the circle

OC = OD = radius = 20 cm.

∴ Triangle OCD is an equilateral triangle.

To find the length of the minor arc CD.

Let s = minor arc CD.

The arc CD subtends 60° at the centre.

θ = 60°

θ = 60° × \(\frac{\pi}{180}\) radians.

θ = \(\frac{\pi}{3}\) radians

We have s = rθ

Question 5.

Find the degree measure of the angle subtended at the centre of the circle of radius 100 cm by an arc of length 22 cm.

Answer:

Given radius r = 100 cm.

Length of arc s = 22 cm.

Angle subtended by the arc at the centre = θ radians

Question 6.

What is the length of the arc intercepted by a central angle of measure 41° in a circle of radius of 10 feet?

Answer:

Central angle subtended by the arc θ = 41°

θ = 41 × \(\frac{\pi}{180}\) Radians

The radius of the circle r = 10 feet

Length of the arc = s

s = rθ

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.

Answer:

Let r_{1} and r_{2} be the radii of the two circles and l be the length of the arc.

Given central angle θ_{1} = 60°

Question 8.

The perimeter of a certain sector of a circle is equal to the length of the arc of a semi-circle having the same radius. Express the angle of the sector in degrees, minutes, and seconds.

Answer:

Let OAB be the sector of a circle of radius r.

The angle of the sector is θ.

Perimeter of the sector = OA + arc AB + OB arc AB = rθ

∴ Perimeter of the sector = r + r θ + r

= 2r + rθ

= r(2 + θ) ———- (1)

Length of the arc of the semi – circle of radius

l = nπ ——– (2)

Given that perimeter the circular sector = Length of the arc of the semi circle of radius r

From equations (1) and (2), we have

r(2 + θ) = πr

2 + θ = π

θ = π – 2

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.

Answer:

Given An airplane, the propeller rotates 1000 times per minute.

∴ A point on the edge of the propeller also rotates 1000 times in 1 minute.

∴ In 1 minute the point describes 1000 × 2π radians angle at the centre.

In 60 seconds the point describes 1000 × 2π radians angle.

∴ In 1 second the angle described = \(\frac{1000 \times 2 \pi}{60}\) radians

Question 10.

A train is moving on a circular track of a 1500 m radius at the rate of 66 km/hr. What angle will it turn in 20 seconds?

Answer:

Radius of the circular track r = 1500 m.

Speed of the train = 66 km/hr

Let θ be the angle made by the path of train at the centre in 20 seconds.

In 1 hr distance moved by train along the circular path = 66 km

In 60 × 60 seconds distance moved = 66 km

∴ In 20 seconds distance moved s = \(\frac{66}{60 \times 60}\) × 20

Question 11.

A circular metallic plate of radius 8 cm and thickness 6 nuns 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.

Answer:

Radius of the circular metallic plate r = 8 cm

Thickness of the plate h = 6 mm = \(\frac{6}{10}\)

Radius of the Pie l = 16 cm

Thickness of the Pie ( h) = 4mm = \(\frac{4}{10}\) cm

Given volume of the cylinder = Volume of the sector