1, Find the quadrants in which the following angles lie :
 i)  45^o         ii)120^o
i) 45^o lies in the first quadrant
        
ii) 120^o lies in the second quadrant
                 

2, Find the quadrants in which the following angles lie:
I, 200^o               ii,300^o
i) 200^o lies in the third quadrant
           
ii) 300^o lies in the fourth quadrant
                

3, Find the quadrants in which the following angles lie.
i, 500
Ans: 500^o = 360^o x 1 + 90^o +50^o
     

4, Find the quadrants in which the following angle lie.
i) 600^o
          600^o = 360^o x 1 + 180^o + 60^o
 
          

5, Find the quadrants in which the following angles lie.
    i) 800^o
800^o = 360^o x 2 + 80^o
          

6, Find the quadrants in which the following angle lie.
   i) -60^o
         

7, Find the quadrants in which the following angle lies.
      i) -120^o
         
 
8, Find the quadrants in which the following angle lie:
         i) -1050^o
    -1050^o = 2 x ( -360^o ) + ( -330^o)
    

9,Find the radian measure corresponding to the following degree measure.
         i) 25^o
   We know that , 180 ^o =  π c
                
Hence, radian measure of 25^o is  

10, Find the radian measure corresponding to the following degree measure.
    i) -47^o 30’
        

11,Find the radian measure corresponding to the following degree measure.
   i) 240^o
We know that  180 ^o = π c
    
Hence, radian measure of 240^o is  

12, Find the radian measure corresponding to the following degree measure.
   i) 520^o
       
Hence, radian measure of 520 ^o is

13, Find the radian measure corresponding to the following degree measure.
    i) ^5π/3
 We know that  180 ^o = π c

   = 300^o
Hence, radian measure of 5π/3  radian os 300^o

14, Find the degree measure corresponding to the following radian measure.
              i) ¹¹/16
                 

15, Find the degree measure corresponding to the following radian measure.of -4^c
          
= - 229^o 5' 27"  approx
Hence  degree measure of -4^c radians is  - 229^o 5' 27"  .

16, Express 5^o 37’ 30” into radians.
           

17, The angles of a triangle are in A.P. The ratio of number of degrees of the least angle, is to the number of radians in the gretest angle is given as 60: π. Find the angles in degrees.
Let the three angles in A.P be a –d, a a +d . Since, the sum of angles of a triangle = 180^o
⇒    a –d, a a + d+ a = 60^o  ⇒
∴ The angles are ( 60 - d ) o, 60^o, ( 60 + d )^o.
Again the number of degree in the least angle = ( 60 – d ) ^o.
And greatest angle = ( 60 – d ) ^o  =  ( 60 + d ) ^{o  π}/ 80  radians
 ∴  By the  given condition ,

         
 
18, The circular measure of two angles of a triangle are ½ radian and 1/3, radian what is the number of degrees in the third angle?
We know that the sum of angles of a triangle is 180^o i.e. π radians. two angles are ½ radian and ¹/3 radians.
         

19, find the angle between the minute hand and the hour – hand of a clock at
      i) 4 : 30 P.M.
We know that the hour-hand completes one rotation in 12 hours whereas the minute – hand completes one rotation in one hour.
i) At 4 : 30 PM., let the hour hand and the minute – hand be along OB and OC respectively.
∴ Required angle = ∠ BOC
= ( Angle subtended by the minute hand in 30 minutes) - (angle subtended by the hour – hand in 4 hours 30 minutes i.e. ⁹/2 hours)
        

20, Find the angle between the minute hand and the hour – hand of a clock at :
At 10: 40 PM., let the hour – hand and the minute – hand be along OC and OB respectively.
Required angle = ∠ BOC
   = ( angle subtended by the hour – hand in 1o hours 40 minutes i.e. 10 ²/3 hours ) – ( angle subtended by the minute – hand in 40 minutes )
          

21, What is positive angle ?
   If the ray rotates OB  about 0 from, the initial position OA in the describes a positive angle. In the anticlockwise direction, we say that it describes a positive angle. In the figure,∠ AOB is a positive angle  ,

For example,
( i )  ∠ AOB  = + 60^o,                 ( ii )  ∠ xoy  = + 135^o
            
 
22, what is negative angle ?
If the ray rotates about 0 from the position OA in the clock wise direction we say that it describes a negative angle. In the figure,∠ AOB  is a negative angle.
 
for example :
( i )  ∠ AOB  = - 30^o,                 ( ii )  ∠ xoy  = - 120^o,
            
 
23, Define degree measure.
In this system, the unit of measuring an angle is a degree. If a right angle is divided in 90 equal parts, then each part is called a degree. One is divided in 60 equal parts, each part is called a minute. One minute is further divided in 60 equal parts, each part is called a second.
1 right angle = 90 degrees ( written as 90^o)
1 degree = 60 minutes ( written as 60’
1 minute = 60 seconds ( written s 60” )
24, Define co – terminal Angles:
Two angles with different measures, but having the same initial sides and the same terminal sides are known as co – terminal angles.

eg: (i) The angles 60^o and – 300^o are co terminal
( ii) The angles, 240^o and -120^o are co terminal
           
25, Define Radian Measure.
In this system the angle subtended the arc at the centre of a circle is measure by dividing the length of the arc by the length of the radius.
          
In this sytem, the angle is measured in radians.

26, Define Radian.
Radian is the angle subtended at the centre of a circle by an arc, whose length is equal to the radius.
              

27, Prove that Radian is a constant Angle.
Let 0 be the centre of a circle of radius r

Now, we have to prove that ∠ AOB is constant. Extend AO to meet the circumference at C. Since Arc subtends∠ AOB at the centre and arc   subtends straight angle AOC.

                   

 
28, Express in circular measures and also in degrees the angle of a regular octagon.
A regular octagon has 8 equal sides
The sum of the 8 exterior angle = 360^o
∴   Each exterior angle = 360^o/8 = 45^o
∴ Each interior angle = 180^o - 45^o = 135^o
Now, we know that 180^o = π radian

      

Hence , the angle of a regular octagon is = 135^o = ^3π/4  radians.

29,Express in circular measure and also in degree the angle of a regular polygon of 40 sides.
Each exterior angle of a regular polygon of 40 sides = 360^o/40 = 9^o
Each interior angle = 180^o – 9^o = 171^o
Now , we know that, 180o =  π radians
   ∴  171^o = ^π /180 x 171  radians = ^19π / 20 radians
Hence the angle of a regular polygon of 40 sides is ^19π / 20.

30. Prove that the number of radians in an angle subtended by an arc of a circle at the centre = arc/ radius.  Ie) θc= 1/r, 

Where θc is the angle subtended at the centre of a circle of radius  r  by an arc of  length l.
Let us consider a circle with centre 0 and radius r. Let =∠ AOB θcand arc AB = l.
Let C be a point on arc AB, Such that arc ac = r
Then ∠ AOB = 1 c
We know from geometry, that the angles at the centre of a circle are proportional to the arcs subtending them .
     

31, Find the length of an arc of a circle radius 3 cm, if the angle subtended at centre is 30o (π= 3.14 )
      Let l be the length of the arc
We know that, Angle  θ = l/r
 
Hence, the arc length is 1.57 cm.

32,In a circle of diameter 40 cm the length of a chord is 20 cm. Find the length of the minor arc of the chord.
We have, Radius ( r ) = ⁴⁰/2 = 20 cm
And length of chord = 20 cm
⇒  Δ OAB is an equilateral triangle

33, Find the angle in radian through which a pendulum swings and its length is 74 cm and the tip describes an arc of length 21cm.

Length of rope = 75 cm
⇒ Radius = 75
Length of the arc = 21 cm
and, we know that angle = ^Arc/ radius
                                    =²¹/ 75 radian
                                    =⁷/25 radians
 
 
 
34, Express 104o 36' in the circular system
        
 
35. The angles of a triangle are in the ratio 2 : 3: 4. Express the angles in circular measwure as well as in degrees.
Let the angles of the triangle be ( 2 x )^o, ( 3 x)^o,  ( 4x) ^o
      ∴ ( 2 x )^o, ( 3 x )^o,  ( 4x) ^o  = 180^o
     ⇒  9x  = 180^o
       ⇒  x = 20
       ∴  The angles are 40^o, 60^o, 80^o
Now   180  = π  radians implies

 
36, Find in radius the angle of a regular octagon
No. of sides = 8
No. of exterior angles = 8
sum of all exterior angles = 360^o
Each exterior angle = 360^o/8 = 45^o
Each interior angle = 180^o – 45^o
                          = 135^o = ( 135 x ^π/180 ) radians
                        = ^3π./4 radians.

37, If in a circle of radius r, an arc of length l subtends an angle θ  radians, then prove that θ = l/ r
    Let o be the centre of the circle of radius r, let arc AB = l and arc AC= r, where units of l and r are same.
Since, angles at the centre of a circle are proportional to the arcs on which they stand, we have
             

38, Find the length of an arc of a circle of radius 10 am which subtends an angle of 45^o at the centre.
Let   ∠ AOB= 45^o,  OA = 10 cm
       arc AB = l cm
              

39, A railway train is travelling on a curve of  500 m radius at the rate of 66 kph.  Through  what angle  will it turn in 20 seconds ?
              

40, In a circle of diameter 40 cm, the length of u chord is 20 cm. Find the length of the minor arc of the chord.
Let o be the centre of the circle of diameter 40 cm.
Let BC be the chord of length 20 cm
   

 
41, If the arcs of the same length in two circles subtend angles 75^o and 120^o at the centre,  Find the ratio of their raddii.
 Let r₁ and r₂ be the radii of the circles.
Let l be the length of each arc.

42. Express angles 60^o, 75^o, 115^o, in the  circular system.