Title XI - maths - chapter 3 - TRIGNOMETRIC RATIOS (1-18).pdf ✅

TRIGONOMETRIC RATIOS 1 JEE MAINS - VOL - II  Angle: An angle is the union of two rays having a common end point in a plane.  Measurement of an angle :  Sexagesimal system: (i) One right angle = 2  radian = 90o. (ii) radian = 2 right angles = 180o. (iii)1 60 minutes(60')   (iv)1' 60seconds(60'')  (v)1o 0.001745  radian (vi) 0 1 11 1radian 57 17 45  (approx)  Centisimal system: (i) 1 right angle = 100 grades written as 100g (ii) 1 grade or 1g = 100 minutes (100’) (iii)1 minute or 1’ = 100 seconds (100”)  Circular system :  Radian: A radian is the angle subtended at the centre of a circle by an arc equal in length to the radius of the circle. The length of arc l r   . (i) 1 revolution = 2 radians 360 (ii)  radians = 2right angles    2 90 180   (iii) 1 degree (1 ) 180    rad  0.01745 rad (iv) 1 rad 180 (1 ) c   degrees  57 17'46"   Note :(i)Value of 22 355 ( ) ( )3.1416 7 113   or or (ii) is an irrational number (iii) circumferenceof thecircle π = diameter of thecircle  Trigonometric Identities : i) sin cos 1, ,      ec n n Z     ii) cos sec 1, 2 1 ,     2 n n Z         iii) tan cot 1, 2 1 ,     2 n n           iv) 2 2 2 2 sin cos 1 sin 1 cos          v) 2 2 2 2 sec tan 1 sec 1 tan          2 1 ,  2 n n Z          . (vi) 2 2 cos cot 1 ec     2 2    cos 1 cot ec       n n Z ,   Note: (i) If 2 1 ,  2 n n Z      , then    2 2 sec tan 1 sec tan sec tan 1 1 sec tan sec tan                     (ii) If     n n Z , then 1 cos cot cos cot ec ec        Trigonometric ratios of various angles: Trig. Ratio 0 30 45 60 90 sin 0 1 2 1 2 3 2 1 cos 1 3 2 1 2 1 2 0 tan 0 1 3 1 3  cos ec  2 2 2 3 1 sec 1 2 3 2 2  cot  3 1 1 3 0 TRIGONOMETRIC RATIOS SYNOPSIS
2 TRIGONOMETRIC RATIOS JEE MAINS - VOL - II 0 or 360 o o 180o 90 o All Positive others negative others negative others negative Sin >0, Cosec   Tan >0, Cot   Cos >0, Sec   Q2 Q3 Q4 Q1 270o Note : i) sin tan cos 2 1 0,   2 n n n n Z          ii)sin 2 1 1 cos 1 ,       2 n n n and n n Z           Domain and range of trigonometric functions : W.E-1: If 4 sin 2 a    then a lies in Sol:    1 sin 1  4 1 1 2 a           2 4 2 a    2 6 a  a 2,6  Some useful results : (a) If A B or   90o 270o, then (i) 2 2 sin sin 1 A B   (ii) 2 2 cos cos 1 A B   (iii)tan . tan 1 A B  (iv) cot .cot 1 A B  (b) If o A B   180 , then (i) cos cos 0 A B   (ii)sin sin 0 A B   (iii)tan tan 0 A B   (c) If A B  360o , then (i) sin sin 0 A B   (ii) cos cos 0 A B   (iii)tan tan 0 A B   W.E-2: tan130o.tan140o  Sol: 130o 140o 270o    tan130o.tan140o  1 W.E-3: 2 2 sin 55o sin 35o   Sol: 55o 35o 90o    2 2 sin 55o sin 35o   1  (i) If a b c cos sin     and a b sin cos    =K then 2 2 2 2 a b c k    (ii) If a sec b tan c     and 2 2 2 2 a tan bsec k then a b c k        (iii) If a cosec bcot c     and 2 2 2 2 a b ec k then a b c k cot cos        W.E-4: If 8cos 6sin 5     then 8sin 6cos     Sol: Let 8sin 6cos    k 2 2 2 2 a b c k    2 2 2 2     8 6 5 k 2      k k 75 5 3  2 0 0 0 0 x if x x x x if x if x            for example the value of 2 cos 100o cos100o     cos100o cos100o 0    (i)sin sin( ) sin(2 ) .......           0 .... sin( ) sin if n is odd n if nis even          (ii) cos cos( ) cos(2 ) .......           0 .... cos( ) cos if n is odd n if nis even         