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Mathematics and Statistics : Part I, , STD : XI, , Angle and its, measurements, Science

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Angle and its measurements, Exercise 1.1, Q.1 A) Determine which of the following pairs of, angles are co-terminal., , Q.1 A) Determine which of the following pairs of, angles are co-terminal., ii) 360°, −3, , Solution :, , i) 210°, −150°, , Two directed angles are co-terminal angles if the, , Solution :, , difference between their measures is an integral, , Two directed angles are co-terminal angles if the, difference between their measures is an integral, multiple 3600, , multiple 3600, Here 360° - (−3°), , = 360° + 30, = 363°, , Here 210° - (−150°) = 210° + 1500, = 360°, ,  210°, and −150° are co-terminal angles., ,  difference between the given angles is not, integral multiple of 3600.,  360°, and −3° are not co-terminal angles., , Ans : 360°, and −3° are not co-terminal angles., Ans : 210°, and −150° are co-terminal angles.

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Angle and its measurements, Q.1 A) Determine which of the following pairs of, angles are co-terminal., iii) -180°, 540°, , Solution :, Two directed angles are co-terminal angles if the, difference between their measures is an integral, , multiple 3600, Here 540° - (−180°) = 540° + 1800, = 720°, = 2 ( 360°),  difference between the given angles is integral, multiple of 3600.,  -180°, and 540° are co-terminal angles., , Ans : -180°, and 540° are co-terminal angles.

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Angle and its measurements, Exercise 1.1, O, , Q.1. B Draw the angles of the following measures, and determine their quadrants., B, , i) –140°, , Solution :, , Y - axis, , , , , , A, - 1400, , Terminal arm of angle lies in the third quadrant, Hence , the angle lies in the third quadrant, , O, , , , , , A, - 1400, , B, , X - axis

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Angle and its measurements, Exercise 1.1, Q.1. B Draw the angles of the following measures, and determine their quadrants., , 2500, , O, , , , , , A, , B, , i) 250°, , Solution :, , Y - axis, , Terminal arm of angle lies in the third quadrant, Hence , the angle lies in the third quadrant, , 2500, , B, , O, , , , , , A, , X - axis

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Angle and its measurements, , B, , , , Exercise 1.1, Q.1. B Draw the angles of the following measures, and determine their quadrants., , 4200, , O, , , , , , A, , i) 420°, , Solution :, , Y - axis, , Terminal arm of angle lies in the first quadrant, Hence , the angle lies in the first quadrant, , , , B, , 4200, , O, , , , , , A, , X - axis

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Angle and its measurements, Q.2 Convert the following angles into radian., , Q.2 Convert the following angles into radian., iii) - 132°, , i) 85°, , Solution :, , Solution :, , θ0  θ, , , , , π , , 180 , , 17, πc, 85 , 180, 36, ,  85 0 , ,  17 , ,  85 0 , , , θ  θ, , 0, , π , , , 180 , , 11, 33, 66,  - 132  - 132 , 0, , πc, 36, , 17 π c, 36, , , , Ans : 85 0, , c, , 17 π c, 36, , 17π c, , 36, , c, , πc, 180, 90, 45, 15, , πc,  - 11 , 15, c, - 11 π,  - 132 0 , 15, , Ans :, , - 11π c, - 132 , 15, 0

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Angle and its measurements, Exercise 1.1, Q.2 Convert the following angles into radian., i) 65° 30’, , Solution :, , , , Consider , 65° 30’ = 65° + 30’, 30 , = 65° + , ,  60 , , 1 , = 65° + , ,  2 , ,  130  1 ,  , ,  2 ,  131 ,  , ,  2 , , , Now θ   θ, , 0, , π, , 180, , 0, , , , , , c, , 0, , 0, , 0, , πc, 131, , 180, 2, , 0, ,  131 ,  ,  ,  2 , ,  65° 30’, , 131π c, 360, , , , 131π c, 360, , Ans : 65° 30’, , , , 131π c, 360

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Angle and its measurements, Exercise 1.1, Q.2 Convert the following angles into radian., i) 40° 48’, , Solution :, Consider , 40° 48’ = 40° + 48’, 48 , = 40° + , ,  60 , , 0, , 4 , = 40° + , ,  5 ,  200  4 ,  , , 5, , ,  244 ,  , ,  5 , , Now θ 0   θ, , , , , 0, , π , , 180 , , c, , 61, 122, 0, πc, 244,  244 , ,  , , , 180, 5,  5 , 90, 45, 61π c, , 5  45, ,  40° 48’, , , , 61π c, 225, , 0, , 0, , Ans : 65° 30’, , , , 131π c, 360

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Angle and its measurements, Exercise 1.1, Q.3 Convert the following angles in degree., (i), , 7π c, 12, , (ii), , Solution :, , , , 15, 7π, 7π, 180, ,  , , , 12, π ,  12, 1, , 180 , , we have , θ c   θ , , π , , , 0, , c, , 0, , , , - 5π, 3, , c, , 60, 5π, 180, , , , ,  , π ,  3, 1, , = (7  15 )0, , = (-5  60 )0, , = 1050, , = - 3000, , c, , 7π,  1050, 12, , Ans :, , - 5π c, 3, , Solution :, , 180 , , we have , θ c   θ , , π , , , , , Q.3 Convert the following angles in degree., , 7π c,  105 0, 12, , , , Ans :, , - 5π c,  - 3000, 3, , - 5π c,  - 300 0, 3, , 0, , 0

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Angle and its measurements, Exercise 1.1, Q.4 Express the following angles in degree, minute and, second., i) (183.7)0, , Solution :, , Q.4 Express the following angles in degree, minute and, second., ii) (245.33)0, , Solution :, , (183.7)0 = 1830 + 0.70, , (245.33)0 =, , 2450 +0. 330, , = 1830 + (0.7  60)’, = 1830 + 42’, , = 2450 + (0. 33  60 )’, , = 1830 , 42’, , = 2450 + 19’ + 0.80’, , Ans : (183.7)0 = 1830 , 42’, , = 2450 + (19.80)’, , = 2450 + 19’ + ( 0.80  60 )’’, = 2450 + 19’ + 48’’, = 2450 , 19’ , 48’’, , Ans : (245.33)0 = 2450 , 19’ , 48’’

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Angle and its measurements, Exercise 1.1, 7π c, Q. 5. In  ABC , mA =, m B = 1200 , find, 36, , m C in degree and radians, ,   C = 250, 5, , Also, , 250, , =, , Solution :, mA =, , 7π c, 36, , 5, 180 ,  7π,  , , , π ,  36, 1, , , 0, , = 350, , m  A + m  B +  C = 1800,  350 + 1200 +  C = 1800,  1550 +  C = 1800, ,   C = 1800 - 1550, , Ans :  C = 250, , πc, 25 , 180, 36, c, 5π, 36, , or, , 5π c, 36

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Angle and its measurements, Exercise 1.1, Q.8. The sum of two angles is 5πc and their difference, is 60°. Find their measures in, , Solution :, 5π c, , , , Substituter x = 4800 in eq. (1), 180 , ,  5π , , π , , , 0, , = 900 0, , x + y = 9000 . . . . . (1), x - y = 600, , . . . . . (2), , Adding eq. (1) and (2) we get, 2 x = 9600,  x , ,  4800 + y = 9000, ,  y = 9000 - 4800,  y = 4200, , Let the measures of two angles be x and y, , 480, 960 0, 2, 1, ,  x = 4800, , Ans : Measure of two angles are 4800 and 4200

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Angle and its measurements, Exercise 1.1, Q.9 The measures of the angles of a triangle are in, , 7k = 7(100 ) = 700, , the ratio 3:7:8. Find their measures in degree, , 8k = 8(100 ) = 800, , and radian., , 1, , Also, , Solution :, , 300, , = 30 , , The ratio of measures of angles is 3 : 7 : 8, 700, , By prperty of triangle, 3k + 7k + 8k = 1800,  18k = 1800, 180,  k , 18, = 100, ,  3k = 3(100 ) = 300, , , , πc, 6, , 7, ,  The degree measures of angles are, ,  k, , πc, 180, 6, , 800, , 7π c, πc, , = 70 , 18, 180, 18, 4, 4π c, πc, , , = 80, 9, 180, 9, , Ans : Measure of angles of triangle are 300 ,700 , 800, and, , πc, 6, , ,, , 7π c, 18, , ,, , 4π c, 9

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Angle and its measurements, Q.10 The measures of the angles of a triangle are, in A.P. and the greatest is 5 times the smallest, (least). Find the angles in degree and radian., , Solution :, The angles of the triangle are in A.P., , Let the angles of the triangle in degrees be, a−d, a, a+d, , 300 0, 6, ,  d , , = 500, , a − d = 600 − 500 = 100, 600 + 500 = 1100, , a +d =, , 1, , Also, , 100, , = 10 , , By property of triangle,, a−d + a + a+d, ,  3a,  a , ,  60 + d = 300 – 5d,  6d = 300, , , , πc, 18, , , , πc, 3, , 1, , =, , 1800, , 600, , =, , = 1800, , πc, 60 , 180, 3, 11, , 180 0, = 600, 3, , 1100, , 11π c, πc, , 110 , 18, 180, 18, , =, , Also , a + d = 5 ( a –d),  60 + d = 5 ( 60 –d), , πc, 180, 18, , Ans : Measure of angles of triangle are 100,600 ,1100, and, , πc, 18, , ,, , πc, 3, , ,, , 11π c, 18

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Angle and its measurements, Q.11 In a cyclic quadrilateral two adjacent angles, c, are 40° and π3 . Find the angles of the, quadrilateral in degree., , Solution :, we know that, , πc, 3, ,   A +  C = 1800,  400 +  C = 1800,   C = 1800 - 400, , = 600, ,   C = 1400, Similarly ,, , D, , A, , Opposite angles of cyclic quadrilateral are, supplementary, ,   B +  D = 1800, C, , B, Let  ABCD be cyclic quadrilateral, Let  A = 400 and  B = 600, ,  600 +  D= 1800,   D = 1800 - 600,   D = 1200, , Ans : Measure of angles of cyclic quadrilateral are, 400, 600 ,1400 and 1200

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Angle and its measurements, Polygon :, The polygon is a flat or plane , two dimensional, closed shape with straight sides., , Regular polygon :, The polygon having all sides equal are called, regular polygon., Measures of all angles of regular polygon are, also equal, , Remember :, , Measure of interior and exterior angle of regular, polygon is given by, 1. m ( Interior angle ) , , 360 0, Number of sides, , 2. m(Interior angle)+m (Exterior angle) =, , 1800, , No. of sides, , Name, , 1, , 3, , Triangle, , 2, , 4, , Quadrilateral, , 3, , 5, , Pentagon, , 4, , 6, , Hexagon, , 5, , 7, , 6, , 8, , Heptagon, Septagon, Octagon, , 7, , 9, , Nonagon, , 8, , 10, , Decagon, , 9, , 12, , Polygon, , 10, , 13, , Polygon

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Angle and its measurements, Exercise 1.1, , 2, 4, , Q.13 Find the degree and radian measure of, , exterior and interior angle of a regular, , 720, , = 72 , , pentagon, 3, 6, , Solution :, Number of sides of regular pentagon = 5, m ( Interior angle ) , , πc, , 180, 10, 5, , 360 0, Number of sides, , 1080, , πc, = 108 , 180, 10, 5, , , , 2π c, 5, , 3π c, 5, , 72, , m ( Interior angle ) , , 360 0, 5, , Ans : For regular pentagon, Measure of interior angle, , 1, ,  Measure of interior angle =, , 720, ,  Measure of exterior angle = 1800 - 720, = 1080, , 720 or, , Measure of exterior angle 1080 or, , 2π c, 5, 3π c, 5

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Angle and its measurements, Exercise 1.1, Q.14 Find the angle between hour-hand and minutehand in a clock at ten past eleven, , Solution :, , At ten past eleven ,, , The angle between minute hand and hour hand, is less than 900.,  , in one minute, hour hand turns to  1 , 2, , 0, , 0, ,  in ten minute , hour hand turns to  1  10  = 50, 2, ,  The angle between hour hand and minute hand, at ten past eleven = 900 - 50, = 850, , When a hour hand moves from one clock, mark to the next one, it turns through an, 0, angle of 360 = 300, 12, , Ans : The angle between hour hand and minute, hand at ten past eleven is 850

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Angle and its measurements, Exercise 1.1, Q.14 Find the angle between hour-hand and minutehand in a clock at twenty past seven, , Solution :, , At twenty past seven ,, , The angle between minute hand and hour hand, is more than 900.,  , in one minute, hour hand turns to  1 , 2, , 0, , 0, ,  in twenty minute ,hour hand turns to  1  20  = 10, 2, ,  The angle between hour hand and minute hand, at ten past eleven = 900 +100, = 1000, , When a hour hand moves from one clock, mark to the next one, it turns through an, 0, angle of 360 = 300, 12, , Ans : The angle between hour hand and minute, hand at twenty past seven is 1000

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Angle and its measurements, Exercise 1.1, Q.14 Find the angle between hour-hand and minutehand in a clock at quarter to six, , Solution :, , At quarter to six ,, , The angle between minute hand and hour hand, is more than 900.,  , in one minute, hour hand turns to  1 , 2, , 0, , 0, ,  in 15 minute ,hour hand turns to  1  15  = 7.50, 2, ,  The angle between hour hand and minute hand, at ten past eleven = 900 +7.50, = 97.50, , = 97° + 0.50, When a hour hand moves from one clock, mark to the next one, it turns through an, 0, angle of 360 = 300, 12, , = 97° + (0.5  60)’, = 97° + 30’, = 97° , 30’, , Ans : The angle between hour hand and minute, hand at quarter to six is 970 , 30’

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Thanks