Title: Unveiling the Basics of Trigonometry I. Introduction Definition of Trigonometry Historical context and origins Importance in mathematics and real-world applications II. Fundamental Concepts Definition of angles and their measurement Introduction to right-angled triangles Primary trigonometric ratios: sine, cosine, tangent III. Trigonometric Functions Definition of trigonometric functions Graphs of sine, cosine, and tangent functions Periodicity and amplitude IV. Trigonometric Identities Pythagorean identity Reciprocal identities Quotient identities V. Solving Triangles Use of trigonometric ratios to solve triangles Application of the Law of Sines and Law of Cosines Examples and practical problem-solving VI. Applications of Trigonometry Navigation and astronomy Engineering and physics applications Everyday scenarios demonstrating trigonometric principles VII. Advanced Topics (Brief Overview) Unit circle and radian measure Trigonometric equations Trigonometric functions of any angle VIII. Interactive Examples and Demonstrations PowerPoint slides demonstrating key concepts Interactive activities for audience engagement Real-life scenarios illustrating trigonometric principles IX. Practical Tips and Tricks Memory aids for trigonometric ratios Problem-solving strategies Common mistakes to avoid X. Conclusion Recap of key concepts Emphasis on the practical relevance of trigonometry Encouragement for further exploration and learning

1. TRIGONOMTERY
Basic Trigonometry

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S. No. TOPICS
1. Logarithms
2. Inequalities - Wavy Curve
3. Trigonometry
4. Functions: Basics, Domain & Range
5. Modulus & Greatest Integer Functions
6. Complex Numbers
7. Differentiation
8. Limits
9. Quadratic Equations
10. Permutations & Combinations
S. No. TOPICS
11. Probability
12. Binomial Theorem
13. Integration
14. Sequence & Series
15. Vectors
16. 3D Geometry
17. Graphs & Transformation
18. Straight Lines & Circles
19. Sets & Relations
20. Conic Sections
21. Matrices and Determinants

3. TRIGONOMETRY
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4. TRIGONOMTERY
● Measurement of an Angle
● Relation between radians and degrees
● Trigonometric Ratios
● Reciprocal and Co-ratio of Trigonometric
Ratios
● Sign of Trigonometric Functions
● Quadrant Angles and Allied Angles
● 3- STEPS to get any ANGLE
● Trigonometric functions of particular angles
● Trigonometric Identities
● Trigonometric Ratios of Negative Angles
● Compound Angles
● Double Angle & Triple Angle Formula
TRIGONOMETRY

5. TRIGONOMTERY
JEE(MAIN) PYQs Using TRIGONOMETRY BASICS

6. TRIGONOMTERY
JEE(MAIN) PYQs Using TRIGONOMETRY BASICS

7. TRIGONOMTERY
Angle Measurements
The amount of rotation of a ‘moving ray' (terminating ray) with
reference to a fixed ray' (initial ray) is called an angle. And it is
denoted by θ or α or β etc.
θ

8. TRIGONOMTERY
Positive Angle
If the rotation of the terminating ray is
in anti-clockwise direction, the angle is
called as positive.
Negative Angle
If the rotation of the terminating ray is
in clockwise direction, the angle is
called as negative.

9. TRIGONOMTERY
System of Measurement of an Angle
(i) The Sexagesimal Measurement
(ii) The circular Measurement

10. TRIGONOMTERY
Definition
1. Sexagesimal system (British System)
It is denoted by 10
If the central angle is divided into 360 equal
parts, each part in it is called One degree.

11. TRIGONOMTERY
Minute
10 is divided into 60 equal parts each part
in it called One minute.
It is denoted as 1′. 10 = 60′
Second
Again if 1’ is divided into 60 equal parts
each part in it called One second.
It is denoted as 1’’. 1’ = 60’’

12. TRIGONOMTERY
Important Points

13. TRIGONOMTERY
Definition
A radian is an angle subtended at the
centre of a circle by an arc whose length is
equal to the radius of the circle.
r
B
A
r
r
1c
2. Circular system (or) Radian Measure

14. TRIGONOMTERY
One radian is denoted as 1c.
Definition
Angle subtended at the centre of a circle of
radius r by an arc of length l is defined as
radians.
l
r
θ =

15. TRIGONOMTERY
Relation between radians and degrees
∴ 1c =
1800
π
1c ≈ (57.272…)0
3600
= 2πc
1c =
(180)0
22
7
=
0
11
630
90
11

16. TRIGONOMTERY
Relation between radians and degrees
Remember
➢ To convert radians into degree
multiply with
➢ To convert degrees into radians
multiply with
π
1800
1800
π

17. TRIGONOMTERY
Q. Express the following angles in
degrees.

18. TRIGONOMTERY
Solution:

19. TRIGONOMTERY
Q. Express the following angles in
degrees.
2
9 π

20. TRIGONOMTERY
Solution:

21. TRIGONOMTERY
Q. Express the following angles in
degrees.

22. TRIGONOMTERY
Solution:

23. TRIGONOMTERY
Q. Determine how the angle radians are
classified.
A
B
C
D
Acute
Right
Obtuse
Reflex
7
8 π

24. TRIGONOMTERY
Solution:

25. TRIGONOMTERY
A
B
C
D
Acute
Right
Obtuse
Reflex
Q. Determine how the angle radians are
classified.
7
8 π

26. TRIGONOMTERY
Q. Convert each degree measure to radian
measure. a. 135° b. 40°

27. TRIGONOMTERY

28. TRIGONOMTERY
Q. Convert 7°30' into radians

29. TRIGONOMTERY

30. TRIGONOMTERY
Trigonometric Ratios
● Sinθ =
Opposite side
Hypotenuse
● Cosθ =
Adjacent side
Hypotenuse
● Tanθ =
Opposite side
Adjacent side
● Cosecθ =
Hypotenuse
Opposite side
● Secθ =
Hypotenuse
Adjacent side
● Cotθ =
Adjacent side
Opposite side

31. TRIGONOMTERY
Reciprocal and Co-ratio of Trigonometric Ratios

32. TRIGONOMTERY

33. TRIGONOMTERY
TRIGONOMETRIC IDENTITIES
1. sin2θ + cos2θ = 1
2. sec2θ – tan2θ = 1
3. cosec2θ – cot2θ = 1

34. TRIGONOMTERY
Values Of T-Ratios At Particular Angles

35. TRIGONOMTERY
sinθ
cosθ
tanθ
00
0
1
0
1
2
1
√2
2
√3 1
√2
1
√3
1
300 450
1
2
√3
0
Undefined
900
2
√3
600
1
0
π
6
π
4
π
3
π
2
T-ratio
Angle (θ)
Trigonometric functions of particular angles

36. TRIGONOMTERY
tanθ 0 0
T-ratio
Angle (θ)
sinθ
cosθ
1800 3600
π 2π
–1
0
1
0
Trigonometric functions of particular angles

37. TRIGONOMTERY
Q.

38. TRIGONOMTERY

39. TRIGONOMTERY
Sign of Trigonometric Functions
I
II
III IV
Y
X

40. TRIGONOMTERY
cosθ < 0 ; sinθ < 0
cosθ < 0 ; sinθ > 0 cosθ > 0 ; sinθ > 0
cosθ > 0 ; sinθ < 0
I
II
III IV
x
y
x > 0, y > 0
x < 0, y > 0
x < 0, y < 0 x > 0, y < 0
Sign of Trigonometric Functions

41. TRIGONOMTERY
Trigonometric Ratios of Negative Angles
sin(-θ) -sinθ
cos(-θ) cosθ
tan(-θ) -tanθ
cot(-θ) -cotθ
sec(-θ) secθ
cosec(-θ) -cosecθ

42. TRIGONOMTERY
Quadrant Angles and Allied Angles

43. TRIGONOMTERY
STEPS to get any ANGLE

44. TRIGONOMTERY
Q. Find the value of the following
trigonometric ratios.
(i) sin(1500) (ii) cos(-2100)

45. TRIGONOMTERY

46. TRIGONOMTERY
Trigonometric Ratios Of Compound Angles

47. TRIGONOMTERY
Sine And Cosine Of Compound Angles
Sum of any two or more angles is called
compound angle.
1) sin (α + β)
3) cos (α + β)
sinα cosβ + cosα sinβ
=
= cosα cosβ – sin α sin β
sin (α – β)
2) = sinα cosβ – cosα sinβ
cos (α – β) = cos α cos β + sin α sin β.
4)

48. TRIGONOMTERY
Sine And Cosine Of Compound Angles
3) cos (α + β) = cosα cosβ – sin α sin β
sin (α – β)
2) = sinα cosβ – cosα sinβ
cos (α – β) = cos α cos β + sin α sin β.
4)

49. TRIGONOMTERY
tan (α + β) =
tan α + tan β
1 – tan α tan β
tan (α – β) =
tan α – tan β
1 + tan α tan β
=
sin (α + β)
cos (α + β)
=
sin (α – β)
cos (α – β)
Similarly,

50. TRIGONOMTERY
Q. Find the value of sin 150

51. TRIGONOMTERY
sin 150
=
=
√3
2
–
1
√2
1
2
=
√3 – 1
2√2
1
√2
sin (450 – 300)
= sin 450 cos300
. – cos450 . sin300
Solution:

52. TRIGONOMTERY
Q. Find the value of cos⁡
75∘.

53. TRIGONOMTERY

54. TRIGONOMTERY
A
B
D
C
-1
-2
1
2
Q. The Value of
is

55. TRIGONOMTERY
Solution:

56. TRIGONOMTERY
A
B
D
C
-1
-2
1
2
Q. The Value of
is

57. TRIGONOMTERY
Double Angle

58. TRIGONOMTERY
Trigonometric functions of Double Angles
I. sin 2θ = 2 sin θ . cos θ
Proof:

59. TRIGONOMTERY
I. sin 2θ = 2 sin θ . cos θ
sin 2θ = sin (θ + θ)
= sin θ . cos θ + cos θ . sin θ
= 2 sin θ . cos θ
Trigonometric functions of Double Angles

60. TRIGONOMTERY
II. cos 2θ = cos2 θ – sin2 θ
= 1 – 2 sin2 θ = 2 cos2 θ – 1
Proof:
Trigonometric functions of Double Angles

61. TRIGONOMTERY
II. cos 2θ = cos2 θ – sin2 θ
= 1 – 2 sin2 θ = 2 cos2 θ – 1
cos 2θ = cos (θ + θ)
= cos θ . cos θ – sin θ . sin θ
∴ cos 2θ = cos2 θ – sin2 θ …. (i)
= (1 – sin2 θ) – sin2 θ
…. (ii)
= 1 – 2 sin2 θ
Also,
cos 2θ = cos2 θ – sin2 θ
= cos2 θ – (1 – cos2 θ)
= cos2 θ – 1 + cos2 θ
= 2 cos2 θ – 1 …. (iii)
Trigonometric functions of Double Angles

62. TRIGONOMTERY
Q. Given equation is equivalent to
A
B
D
C
cos 2x
sin 2x
sin x
cos x

63. TRIGONOMTERY
Solution:

64. TRIGONOMTERY
Q. Given equation is equivalent to
A
B
D
C
cos 2x
sin 2x
sin x
cos x

65. TRIGONOMTERY
III. tan 2θ =
2 tan θ
1 – tan2 θ
Proof:
Trigonometric functions of Double Angles

66. TRIGONOMTERY
III.
tan 2θ
tan 2θ =
2 tan θ
1 – tan2 θ
= tan (θ + θ)
=
tan θ + tan θ
1 – tan θ . tan θ
∴ tan 2θ =
2 tan θ
1 – tan2 θ
Trigonometric functions of Double Angles

67. TRIGONOMTERY
Q.Prove: 2 tan x
1 + tan2 x
i) sin 2x =
ii) cos 2x =
1– tan2 x
1 + tan2 x

68. TRIGONOMTERY
i) R.H.S=
2 sin x / cos x
1 +
sin2 x
cos2 x
2 tan x
1 + tan2 x
=
=
2 sin x
cos x
×
(cos2 x + sin2
x)
cos2
x
2 sin x cos x
= sin 2x = L.H.S
=
Solution:

69. TRIGONOMTERY
ii) R.H.S =
1– tan2 x
1 + tan2 x
1 –
sin2 x
cos2 x
1 +
sin2 x
cos2 x
= =
cos2 x − sin2 x
cos2 x
cos2 x + sin2 x
cos2 x
=
cos2 x – sin2 x
cos2 x + sin2 x
= cos 2x = L.H.S
Solution:

70. TRIGONOMTERY
Q.

71. TRIGONOMTERY

72. TRIGONOMTERY
Triple Angle Formulae

73. TRIGONOMTERY
Triple Angle Formulae
●
●
●

74. TRIGONOMTERY
Q. If sinA= 3/4 then value of sin3A will be:
A
B
D
C
9/16
-9/16
9/32
7/16

75. TRIGONOMTERY

76. TRIGONOMTERY
Q. If sinA= 3/4 then value of sin3A will be:
A
B
D
C
9/16
-9/16
9/32
7/16

77. TRIGONOMTERY
C and D Formulae

78. TRIGONOMTERY
Q. Express sin6θ – sin2θ as a product.

79. TRIGONOMTERY

80. TRIGONOMTERY
Defactorization Formulae
(i) 2sinAcosB = sin(A+B) + sin(A–B)
(ii) 2cosAsinB = sin(A+B) – sin(A–B)
(iii) 2cosAcosB = cos(A+B) + cos(A–B)
(iv) –2sinAsinB = cos(A+B) – cos(A–B)

81. TRIGONOMTERY
Q. Express 2 Cos7x Cos3x as a Sum.

82. TRIGONOMTERY