are its positions as it is moving in anti-clockwise direction through angles 1, 2, 3 &

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1 T: Introduction: The word trigonometry is derived from Greek words trigon meaning a triangle and metron meaning measurement. In this branch of mathematics, we study relationship of sides and angles of triangle. T: ngle: It is defined as the amount of turn or rotation of a moving line with respect to a fixed line and fixed point. Y X X Y Here OX is a fixed line and O is a moving line.,, & 4 are its positions as it is moving in anti-clockwise direction through angles,, & 4.When it reaches again at point, it completes one revolution and it is said to have described an angle of 60. T: Positive and Negative angle: When line O is moving in anti-clockwise direction, then the angle described by it is said to positive. But when it moves in clockwise direction the angle made by it is said to be negative. T4 Degrees Measure of an angle: circle is divided into 60 equal parts and the angle subtended by each part is known as. Two perpendicular diameters of a circle divide it into 4 equal parts; therefore, each part subtends an angle of 90 at the centre. right angle 90 (read as 90 ) 60 (read as 60 minutes) 60 (read as 60 seconds) T5: Naming a Right angle: Side Opp. Right ngle Hypotenuse Side Opp to ngle Perpendicular Side djacent to ngle Base 4 B C T6: Property of Similar Triangles:

2 P B C Q R If BC PQR then: B BC C PQ QR PR P, B Q & C R T7: Trigonometric Ratios: Consider a right angled triangle BC and PQ in which B 90 & Q 90 &. P C Q B BC PQ Perpendicular SideOppositeto angle sin C P Hypotenuse Hypotenuse B Q Base Side djacent to cos C P Hypotenuse Hypotenuse BC PQ Perpendicular SideOppositeto tan B Q Hypotenuse Side djacent to C P Hypotenuse Hypotenuse cos ec BC PQ Perpendicular SideOppositeto angle sin C P Hypotenuse Hypotenuse sec B Q Base Side djacent to cos B Q Hypotenuse Side djacent to cos cot BC PQ Perpendicular SideOppositeto tan sin

3 T8 Trigonometrical Identities: sin cos cos sin sin cos sec tan sec tan tan sec cos ec cot cos ec cot cot cos ec T9 Trigonometric Ratios of Complementary ngles: sin(90 ) cos cos ec(90 ) sec cos(90 ) sin sec(90 ) cosec tan(90 ) cot cot(90 ) tan T0 Trigonometric Table: ngle T Ratio 0 sin 0 cos 0 tan 0 cosec Not defined sec 45 cot Not defined Not defined Not defined 0 Trigonometric Identities (Questions) Simplify:. (sec tan )( sin ). sin (cosec cot ). cot tan

4 sin cos 4. sin cos sin cos sin cos 5 5tan 8. cot tan 9. cosec sec 5cosec 5sec 0. cot tan Using trigonometric identities, write the following expressions as an integer:. 4tan 4sec. cot cos ec. 8sin 8cos 4. 6sec 6 tan 5. 7cosec 7cot 5 Prove the following identies cos tan 6. tan sin tan cos ec 7. tan sec cosec sin cos sec sec 8. cosec sec sec cos sin 9. cot tan sin cos sec sin 0. cosec cos tan cot. cos ec cot cot 4 4. sin cos sin cos 6 6. sin cos sin cos sin 4. sec cos tan tan 4 sec sec 4

5 sin 5. sec tan sin sin cos 6. sin sin cos sin 7. cos cos 8. sin cos sec cos sin sin cos sin cos 9. sin cos sin cos sin 0. (sec cos )(cosec sin ) tan cot cos. cot tan sin cos sec tan sin. tan sec cos cot cot. tan sec tan cot 4. tan cot cot tan tan tan 5. cosec sec sec 6. ( tan )( sin )( sin ) sin 7. tan cot sin cos 8. (sin sec ) (cos cosec ) ( sec cosec ) sec sec cosec cosec cot tan sin cos 40. cosec cos sin sin sin 4. cot cosec cot cosec 4. tan cot sec cosec 4. cosec cot sin sin cos ec cot cos sin 44. sincos tan sin cos cos sin 45. cos sin tan cot 5

6 46. ( tan tan ) (tan tan ) sec sec B B B sec sin 47. cot sec 0 sin sec 48. If tan sin mand tan sin m; prove that m n 4 mn 49. If cot cos and cot cos ;prove that 4 a b b a ab 50. If cos sin cos,show that cos sin sin 5. If sin cos p and sec cosec qshow that q( p ) p. cos cos 5. If mand n,show that ( n m ) cos n cos sin 5. If x r sin cos C, y r sinsin C & z r cos,prove that r x y z 54. Prove that ( sin cos ) ( cos )( sin ) 55. If 4 sin sin, prove that cos cos ( sin ) ( sin ) sin 56. cos sin cos cos 57. tan cosec cosec sin sin 58. cot sec sec 59. If asec b tan m & a tan bsec n,prove that m n a b Express in terms of Trigonometrical ratio of angles between 0 & 45:. sin 7 tan 7. cos85 cot 6. tan 57 cot 75 Evaluate:. sin 0 cos 70. tan 4 cot 48. sec80 cosec0 4. sin 55 cos 5 5. tan 8 cot 8 6. cosec 6 sec sin 50 cos 40 cos 0 8. sin 50 sin cos 40 cos 50 sin 8 sin 5 6

7 sin 5 sec55 0. cos 55 cosec5. sec 7 cot 7. cosec 0 tan 70 sin(90 ) sin. cos cos(90 ) 4. cosec tan (90 ) 5. cot sec (90 ) cos 0 cos 6. sin 70 sin(90 ) 7. sin cos(90 ) cos sin(90 ) 8. sin sin(90 ) cos cos(90 ) 9. cot(90 ) sin(90 ) cos(90 ) cos 0. cos (90 ) tan tan. sin(90 ) cos(90 ) tan sin 5 cosec. cos8 sec58. tan5 tan 0 tan 70 tan tan5 tan 0 tan 45 tan 70 tan cot0cot 5cot 65cot80 tan 5 cot cot 7 tan0 7. sin 0 sin 70 tan 45 cos70 cos59 8. sin 0 sin 0 sin sin 0 9. sin 70 sin(90 ) sin cos(90 ) cos 0 cos 70 tan cot cos 40 cos cos(40 ) sin(50 ) sin 40 sin 50 cos70 cos55cosec5. sin 0 tan 5 tan 5 tan 45 tan 65 tan (cos 0 sin 45 ) (sin 60 sec 45 ) cot (sin 0 cos 60 ) (sin 60 cos 45 ) tan Find,if sin( 5 ) cos ; where( 6 ) is acute. 7

8 5. If sec cosec( 6 ), where & ( 6 ) both are acute; find the value of. 6. If tan cot( 6 ), where & 6 are acute, find the value of. 7. If are acute angles and tan cot B; prove that B 90 sec 8. If B 90 ; show that cos sin B& cosec B 9. If B C (ii) tan B C cot C B (iii) sec cos ec 40. If sin cosec ; show that; sin cos ec & B, B & C are the interior angles of a triangle; show that: (i) sin cos 4. If tan cot show that: tan cos 6 8