TOPICS
Exercise - 3.1

Question-1 :-  Find the radian measures corresponding to the following degree measures:
(i) 25°  (ii) – 47°30′  (iii) 240°  (iv) 520°.

Solution :-
(i) 25°
    We know that 180° = π radian.
    Hence,
    25° = π/180 x 25 radian 
        = 5π/36 radian
    Therefore, 25° = 5π/36 radian.

(ii) – 47°30′
    We know that 180° = π radian.
    Hence,
    -47° 30′ = -47 + 1/2 degree 
             = π/180 x (-95/2) radian 
             = -93π/360 radian
             = -31π/120
    Therefore, -47° 30′ = -31π/120 radian

(iii) 240°
    We know that 180° = π radian.
    Hence,
    240° = π/180 x 240 radian 
         = 4π/3 radian
    Therefore, 240° = 4π/3 radian.
  

(iv) 520°
    We know that 180° = π radian.
    Hence,
    520° = π/180 x 520 radian 
         = 26π/9 radian
    Therefore, 540° = 26π/9 radian.
     

Question-2 :-  Find the degree measures corresponding to the following radian measures (Use π = 22/7)
(i) 11/16  (ii) -4  (iii) 5π/3  (iv) 7π/6.

Solution :-
(i) 11/16
  We know that π radian = 180°.
  Hence, 
  11/16 radians = 180/π x 11/16 degree 
            = (45 x 11 x 7)/ (22 x 4) degree 
            = 315/8 degree
            = 39° + 3/8 degree 
            = 39° + (3 x 60)/8 minute    [1° = 60']
            = 39° + 180/8 minute
            = 39° + 22' + 1/2 minute     [1' = 60"]
            = 39° + 22' + 30" 
  11/16 radians = 39° 22' 30".

(ii) -4
  -4 radians = 180/π x (-4) degree 
            = [180 x 7 x (-4)]/22 degree 
            = -2520/11 degree
            = -229° + 1/11 degree 
            = -229° + (1 x 60)/11 minute    [1° = 60']
            = -229° + 60/11 minute
            = -229° + 5' + 5/11 minute     [1' = 60"]
            = -229° + 5' + 27" 
  -4 radians = -229° 5' 27" approximately.

(iii) 5π/3
  5π/3 radians = 180/π x 5π/3 degree 
            = 300° 
  5π/3 radians = 300°.

(iv) 7π/6
  7π/6 radians = 180/π x 7π/6 degree 
            = 210° 
  7π/6 radians = 210°.
   

Question-3 :-  A wheel makes 360 revolutions in one minute. Through how many radians does it turn in one second?

Solution :-
  No. of revolution made by wheel in 1 minute = 360°
  No. of revolution made by wheel in 1 second = 360/60 = 6
  In one complete revolution, the wheels turns an angle of 2π radian.
  Hence, in 6 complete revolutions, it will turn  an angle of 6 x 2π radian. 
  i.e., 12π radian
  Thus, in one second, the wheels 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 (Use π = 22/7).

Solution :-
  We know that in a circle of radius r unit, length of arc l and angle θ. 
  Then, l = r.θ
  Given :
  radius of circle (r) = 100 cm
  length of arc (l) = 22 cm
  angle (θ) = ?
  θ = l/r = 22/100 radian 
          = 180/π x 22/100 degree 
          = (180 x 22 x 7)/ (22 x 100) degree 
          = 126/10 degree
          = 12° + 3/5 degree 
          = 12° + (3 x 60)/5 minute    [1° = 60']
          = 39° + 36' 
          = 39° 36' 
  The required angle is 39° 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.

Solution :-
  Given:
  Diameter of circle = 40 cm
  radius of circle = 40/2 = 20 cm
  length of chord = 20 cm
  length of minor arc of chord = ?
  
  Let AB be a chord(length = 20 cm) of the circle. 
  circle
  In ∆AOB,
  OA = OB = radius of circle = 20 cm
  Also, AB = 20 cm
  Thus, ∆AOB is a an equilateral triangle.
  So, θ = 60° = π/3 radian
  We know that in a circle of radius r unit, length of arc l and angle θ. 
  Then, l = r.θ
  length of arc (AB) = 20 x π/3 = 20π/3 cm
  Length of minor arc of chord is 20π/3 cm.
   

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.

Solution :-
  Let the radii of two circles are r₁ and r₂.
  Let an arc of length l subtend an angle of 60° at the centre of the circle of radius r₁
  Let an arc of length l subtend an angle of 75° at the centre of the circle of radius r₂

  Now, 60° = π/3 radian and 75° = 5π/12 radian
  We know that in a circle of radius r unit, length of arc l and angle θ. 
  Then, l = r.θ
  l = r₁.θ₁ and l = r₂.θ₂
  l = π/3 . r₁ and l = 5π/12 . r₂
  π/3 . r₁ = 5π/12 . r₂
  r₁/r₂ = 5π/12 x 3/π
  r₁/r₂ = 5/4
  r₁ : r₂ = 5 : 4
  Ratio of radii is 5 : 4
    

Question-7 :-  Find the angle in radian 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) 21 cm

Solution :-
  We know that in a circle of radius r unit, length of arc l and angle θ. 
  Then, l = r.θ
  Given:
         
(i) 10 cm 
  length of arc (l) = 10 cm
  radius of circle (r) i.e.,Pendulum = 75 cm
  angle (θ) = ?
  θ = l/r = 10/75 = 2/15 radian

(ii) 15 cm
  length of arc (l) = 15 cm
  radius of circle (r) i.e.,Pendulum = 75 cm
  angle (θ) = ?
  θ = l/r = 15/75 = 1/5 radian

(iii) 21 cm
  length of arc (l) = 21 cm
  radius of circle (r) i.e.,Pendulum = 75 cm
  angle (θ) = ?
  θ = l/r = 21/75 = 7/25 radian
    
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