Tiêu đề XI - maths - chapter 3 - TRIGNOMETRIC RATIOS (1-17).pdf ✅

JEE-MAIN-JR-MATHS VOL-II TRIGONOMETRIC RATIOS NARAYANAGROUP 1  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
TRIGONOMETRIC RATIOS JEE-MAIN-JR-MATHS VOL-II 2 NARAYANAGROUP 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         
JEE-MAIN-JR-MATHS VOL-II TRIGONOMETRIC RATIOS NARAYANAGROUP 3 W.E-5: sin sin ..... sin 100                Sol: n 100 is even  the required value = sin W.E-6: cos cos ..... cos 299                Sol: n  299 is odd  required value = 0  If a>0, b > 0 and f x   0 then     2 b af x ab f x   Proof: We know that A.M.  G.M.       2   b af x f x b af x f x        2 b af x ab f x     Note: (i) 1 x 2 x   , for x > 0 (ii) a x b x ab tan cot 2   (iii) 25cos 16sec 2 25 16 40 x x      (i) 2 2 2 2 cot cos cot cos       (ii) 2 2 2 2 tan sin tan sin       (iii) 2 2 2 2 cos sec cos .sec ec ec        (i) 4 4 2 2 sin cos 1 2sin cos        (ii) 6 6 2 2 sin cos 1 3sin cos        (iii) 2 4 2 2 sin cos 1 sin cos        (iv) 2 4 2 2 cos sin 1 sin cos        W.E-7: 6 6 2 4 2 sin cos sin cos         Sol:   2 2 2 2 2 1 3sin cos 1 sin cos            2 2 2 2 3 1 sin cos 3 1 sin cos          Graph of sin x :  Graph of cos x :  Graph of tan x :  Graph of cot x :
TRIGONOMETRIC RATIOS JEE-MAIN-JR-MATHS VOL-II 4 NARAYANAGROUP 1. 13 / 6  radians = 1)390o 2)- 490o 3) 410o 4) 30o 2. Cot1358o tan 3608o      1) -1 2)0 3)1 4)2 3. sin( 660o ) tan(1050o ) sec( 420o ) cos(225o ) cos ec(315o ) cos(510o )    1) 3 4 2) 3 2 3) 2 3 4) 4 3 4. sin 48o sec 42o cos 48o cos 42o   ec 1) 0 2) 2 3) 1 4) -1 5. lo g ta n 1 7 o lo g ta n 3 7 o     log tan 53o log tan 73o 1) 0 2) 1 3) 2 4) 3 6. cos 24o cos 55o cos125o cos 204o     1) –1 2) 0 3) 1 4) 2 7. sec A tan A 3 sec A     1) 10 3 2) 5 3 3) 2 3 4) 4 3 8. sec tan 3       lies in the quadrant 1) I 2) II 3) III 4) IV 9. sin10o sin 20o sin30o ...... sin360o      1) 0 2) 1 3) – 1 4) 2 10. 4 2 3[sin x cos x] 6[sin x cos x]     6 6 4[sin x cos x]   1) 3 2) 6 3) 4 4) 13 11. 2 2 2 2 2 2 sin tan cos 1 cot (1 tan )           1) – 1 2) 0 3) 1 4) 2 12. 2 1 cos sin 1 1 cos sin 1 cos Sin x x x x x x        1) sin x 2)cos x 3) cosec x 4) secx 13. If A, B, C are the angles of a triangle ABC then 3A 2B C A C cos cos 2 2                  1) 0 2) 1 3) cosA 4) cosC 14. cot cos ec 1 cot cosec 1          1) sin 1 cos    2) sin 1 cos    3) 1 cos sin    4) cos 1 sin    15. 2 2 sin (51o x) sin (39o x)     1) –1 2) 0 3) 1 4) 2 16. 2 2 2 2 2 2 2 2 7 4 sin sin sin sin 18 9 18 9 7 4 cos cos cos cos 18 9 18 9                1) 1 2) 2 3) 3 4) 4 17. If ABCD is a cyclic quadrilateral, then cos(180o A ) cos(180o B)     cos(18 0 o C ) sin (90 o D )     1) –1 2) 0 3) 1 4) 2 18.   2 sin sin 2 x x                     2 3 cos cos 2 2 x x                   1) 0 2) 2 3) 4 4) 8 19. 1 2 3 sin sin sin 3        1 2 3 cos cos cos       1) 0 2) 1 3) 2 4) 3 20. 2 2 8sin x 3cos x 5 cot x     1) 1 2  2) 1 3  3) 3 2  4) 2 3  21. 2 2 4 sin x sin x 1 cos x cos x      1) 0 2) 1 3) 2 4) –1 22. If cos 1  x y     , then cos cos x y   1) 0 2) 1 3) 2 4) –1 23. If 8 8 sin x cosecx 2 then sin x cosec x     1) 1 2) 2 3) 3 4) 4 24. If sin cos x x a   then | sin cos | x x   1) 2 1 a 2) 2 a 1 3) 2 2  a 4) 2 a  2 LEVEL - I (C.W)