## EXERCISE 3.1

#### Page No 54:

#### Question 1:

Find the radian measures corresponding to the following degree measures:

(i) 25° (ii) – 47° 30′ (iii) 240° (iv) 520°

#### Answer:

(i) 25°

We know that 180° = π radian

(ii) –47° 30′

–47° 30′ = degree [1° = 60′]

degree

Since 180° = π radian

(iii) 240°

We know that 180° = π radian

(iv) 520°

We know that 180° = π radian

#### Page No 55:

#### Question 2:

Find the degree measures corresponding to the following radian measures

.

(i) (ii) – 4 (iii) (iv)

#### Answer:

(i)

We know that π radian = 180°

(ii) – 4

We know that π radian = 180°

(iii)

We know that π radian = 180°

(iv)

We know that π radian = 180°

#### Question 3:

A wheel makes 360 revolutions in one minute. Through how many radians does it turn in one second?

#### Answer:

Number of revolutions made by the wheel in 1 minute = 360

∴Number of revolutions made by the wheel in 1 second =

In one complete revolution, the wheel turns an angle of 2π radian.

Hence, in 6 complete revolutions, it will turn an angle of 6 × 2π radian, i.e.,

12 π radian

Thus, in one second, the wheel turns an angle of 12π radian.

#### Question 4:

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

#### Answer:

We know that in a circle of radius *r* unit, if an arc of length *l* unit subtends an angle *θ* radian at the centre, then

Therefore, forr = 100 cm, l = 22 cm, we have

Thus, the required angle is 12°36′.

#### Question 5:

In a circle of diameter 40 cm, the length of a chord is 20 cm. Find the length of minor arc of the chord.

#### Answer:

Diameter of the circle = 40 cm

∴Radius (*r*) of the circle =

Let AB be a chord (length = 20 cm) of the circle.

In ΔOAB, OA = OB = Radius of circle = 20 cm

Also, AB = 20 cm

Thus, ΔOAB is an equilateral triangle.

∴θ = 60° =

We know that in a circle of radius *r* unit, if an arc of length *l* unit subtends an angle *θ* radian at the centre, then.

Thus, the length of the minor arc of the chord is.

#### Question 6:

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 the radii of the two circles be and. Let an arc of length *l* subtend an angle of 60° at the centre of the circle of radius *r*_{1}, while let an arc of length *l *subtend an angle of 75° at the centre of the circle of radius *r*_{2}.

Now, 60° =and 75° =

We know that in a circle of radius *r* unit, if an arc of length *l* unit subtends an angle *θ* radian at the centre, then.

Thus, the ratio of the radii is 5:4.

#### Question 7:

Find the angle in radian though 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) 21 cm

#### Answer:

We know that in a circle of radius *r* unit, if an arc of length *l* unit subtends an angle *θ* radian at the centre, then.

It is given that *r* = 75 cm

(i) Here, *l* = 10 cm

(ii) Here, *l *= 15 cm

(iii) Here, *l *= 21 cm