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1. TRIGONOMETRIC RATIOS For More Details, Click : https://anandclasses.co.in/jee-coaching-center-in-jalandhar/jee-coaching-institute-near-me/

2. TRIGONOMETRIC RATIOS

3. TRIGONOMETRIC RATIOS TRIGONOMETRIC RATIOS

4. INTRODUCTION OF TRIGONOMETRY

5. What is Trigonometry? The word ‘trigonometry’ is derived from the ‘Greek’ words ‘gonia’ means an angle or side ‘tri’ means three ‘metron’ means measure. i) tri ii) gonia iii) metron Hence, trigonometry means science of measuring ‘triangles’.

6. What is an Angle? The amount of rotation of a ‘moving ray’ (terminating ray) with reference to a ‘fixed ray’ (intial ray) is called an ‘angle’. And it is denoted by  or  or  etc. terminal ray  InItial ray

7. Positive Angle If the rotation of the terminating ray is in anti clock wise direction, the angle is regarded as positive. terminal ray  Initial ray

8. Negative Angle If the rotation of the terminating ray is in clock wise direction, the angle is regarded as negative Initial ray - terminal ray

9. Mcqs 1) Trigonometry means science of _______________ 1) angles 2) sides 3) triangles 4) polygons

10. 2) If the rotation of the terminating ray is in anti clock wise direction, the angle is regarded as _________________ 1) Positive 2) Negative 3) Both 4) None of these

11. 3) If the rotation of the terminating ray is in clock wise direction, the angle is regarded as 1) Positive 2) Negative 3) Both 4) None of these

12. MEASUREMENT OF ANGLES

13. MEASUREMENT OF ANGLES How to measure an angle? British system There are three systems to measure angles French system i) The sexagesimal measurement. Radian system ii) The centesimal measurement iii) The circular measurement

14. i) The Sexagesimal measurement (British system) In this system the unit of measurement of an angle is “degree”. What is the definition of a degree? Degree In this system one complete rotation is divided into 360 equal parts. Each part is called ‘a degree’, denoted by 10.

15. Minute A degree is further divided into 60 equal parts and each part is called “one minute”, denoted as 1' Second A minute is further divided into 60 equal parts and each part is called “one second”, denoted as 1"

16. Note 1) of a complete rotation = 10( a degree) 2) of a degree = 1' ( a minute) 3) of a minute = 1" ( a second) 4) One right angle = 900

17. ii) The Centesimal Measurement (French system): “grade”. In this system the unit of measurement of an angle is Grade What is the definition of a grade? In this system one complete rotation is divided onto 400 equal parts. Each part is called “a grade” denoted by 1g.

18. Minute A degree is further divided into 100 equal parts and each part is called “a minute”, denoted as 1' Second A minute is further divided into 100 equal parts and each part is called “a second”, denoted as 1"

19. Note 1) of a complete rotation = 1g( a grade) 2) of a grade = 1' ( a minute) 3) of a minute = 1"( a second) 4) In this system one right angle = 100g

20. iii) The Circular measurement (Radian system):- In this system the unit of measurement of an angle is “radian”. Radian The angle subtended by an arc of length equal to the radius of the circle at its centre is called one radian, denoted by 1c. r O 1c r r=l

21. Note 1) Radian is a constant angle = radian 2) One right angle = right angle 3) One radian 4) If no unit of measurement is indicated for an angle, it will be understood that radian measure is implied

22. Mcqs 1) One minute in the centesimal system =____________ seconds 1) 60 2) 360 3) 400 4) 100

23. 2) In the sexagesimal system one minute =__________ seconds 1) 360 2) 180 3) 90 4) 60

24. 3) The centesimal system is also known as __________ system 1) British 2) French 3) Indian 4) American

25. RELATION AMONG THREE SYSTEMS

26. Relation among the three systems D=Degree G=Grade R=Radian 1) The formula connecting the three systems is as follows, Where D= degree, G= grade, R = radian 2) One complete angle = 3600 = 400g = 2c 3) One straight angle =200g = c = 1800 4) One right angle = 900 =100g =

27. 5) 10 = 0.01745c(approximately) 6) 1c = 570 17' 45" (approximately) 7) 10= 8) 1c=

28. Mcqs 1) 30°=__________ radians Hint: We know that ,

29. 2) 45°=__________ gradians 1) 60 3) 40 4) 30 2) 50 Hint: We know that ,

30. 3) In the sexagesimal system straight angle =___________________ 1) 360° 2) 180° 3) 90° 4) 60°

31. TRIGNOMETRIC RATIOS

32. Trigonometric Ratios The ratios of different pairs of sides of the right angled triangle are called “trigonometric ratios” or “trigonometric functions”.

33. Take an angle of measure  in radian in the standard position. Let P(x,y) be a point on the terminal side of the angle  such that OP=r(>0) Y P(x,y) r y  X X’ O M x Y’

34. Opposite side (o) Hypotenuse (h) Adjacent side (a) SohCah Toa

35. Opposite side Adjacent side Opposite side Opposite side Adjacent side Adjacent side Hypotenuse Hypotenuse Hypotenuse Let us see the ratios of different pairs of sides q With the help of this, let us create 3 ratios Let us consider all three sides of right angled ABC Adjacent side Hypotenuse sin q = yes Can we create more ratios ? A 3 more ratios can be created For q, Opposite side – Adjacent side – cosq = What will be the reciprocal of this ratio ? Hypotenuse Adjacent side Hypotenuse Hypotenuse Consider the measure of A to be q Side BC ? Opposite side Opposite side ? Opposite side Opposite side Adjacent side Side AB Each of these ratios are given a name. They are… What will be the reciprocal of this ratio ? tan q = Hypotenuse So, in all 6 ratios can be created Adjacent side What will be the reciprocal of this ratio? cosec q = Adjacent side Opposite side sec q = B C cot q =

36. Mcqs 1) In right angled triangle tan=_______

37. 2) In right angled triangle cosec=___________

38. 3) In right angled triangle cot=___________

39. QUADRANT ANGLES AND ALLIED ANGLES

40. Quadrant Angles and Allied Angles • 00,900,1800,2700,3600,……are called “quadrant angles” • Two angles are said to be allied when their sum or difference is or a multiple of either zero • etc., are called “allied angles” to

41. Note 1) For 00 , 1800 , 3600 , we get same ratios 2) For 900 , 2700 , we get co-ratios i.e., sin cos, tan  cot, sec cosec

42. Reciprocal and Co-ratios of Trigonometric Ratios Ratio Co-Ratio Reciprocal sin cos cosec cos sin sec tan cot cot cot tan tan sec cosec cos cosec sec sin

43. Mcqs 1) Two angles are said to be allied when their sum or difference is either ________________ 1) zero 3) Both 1 & 2 4)

44. 2) The co-ratio of cos is ___________ 1) sin 2) sec 3) cot 4) cosec

45. 3) Reciprocal of sin is……… 1) sin 2) cos 3) cosec 4) cot

46. SIGNS OF THE TRIGONOMETRIC FUNCTIONS

47. Note:- With the sentence “All Students Take this Chart ” we can remember the signs of trigonometric ratios Signs of Trigonometric Functions tan & Cot sin & Cosec cos & sec sin, cosec cos, sec tan, cot y All Students S Q2 Q1 90° 90°+ 360°+  180°-  90°<  < 180° 0°<  < 90° x’ x O 270°<  < 360° 180°<  < 270° Q4 270°+  180°+  Q3 360° or ( ) 270° Take This T Chart C y’

48. Trigonometric Functions of Allied Angles sin cos -sin cos -sin -cos -cos -sin sin sin cos cos -sin cos cos -cos -cos sin -sin  sin tan -tan -cot -tan -tan tan cot -cot tan cot cot cot -tan tan -cot tan cot -cot -cot -tan sec sec sec sec -cosec cosec -sec -sec cosec -cosec cosec -sec cosec -cosec -cosec sec -sec -cosec sec cosec

49. Trigonometric functions of 2n+ and 2n: Trigonometric functions of 2n+ or 2n are same as 2+ or 2. (n Z) 2n  Q4 2n+  Q1 Sin (2n) =  sin Sin (2n+) = sin Cos (2n ) = cos  Cos (2n+ ) = cos  Tan (2n ) = tan Tan (2n+ ) = tan  Cot (2n ) = cot  Cot (2n+ ) = cot  Sec (2n ) = sec  Sec (2n+ ) = sec  Cosec (2n+ ) = cosec  Cosec (2n ) = cosec

50. Trigonometric Functions of n± We can easily understand the trigonometric functions of n± with the following diagram: Q2 y Q1 n- n+ n is even n is odd x x’ O n+ n Q3 Q4 y’