Trigonometry might sound like a big, complex word, but it’s actually a fascinating branch of mathematics that deals with triangles and the relationships between their sides and angles.
Trigonometry involves understanding the relationships between angles and sides of triangles. It encompasses various concepts, including sine, cosine, tangent, and their inverses. Here are some fundamental terms:
- Sine (sin): Ratio of the length of the side opposite an angle to the length of the hypotenuse in a right triangle.
- Cosine (cos): Ratio of the length of the side adjacent to an angle to the length of the hypotenuse in a right triangle.
- Tangent (tan): Ratio of the length of the side opposite an angle to the length of the side adjacent to the angle in a right triangle.
- Trigonometric identities: Equations that are true for all values of the variables involved, widely used in simplifying trigonometric expressions.
Trigonometry Table:
A trigonometry table lists the values of trigonometric functions for various angles. It’s a handy reference tool for quickly finding sine, cosine, and tangent values for common angles.
Applications of Trigonometry:
Trigonometry finds applications in various fields, including:
- Physics: Used to analyze the motion of objects, such as projectiles and pendulums.
- Engineering: Essential for designing structures like bridges and buildings.
- Astronomy: Helps in understanding celestial bodies’ positions and movements.
- Surveying and Navigation: Used to measure distances and determine directions accurately.
History of Trigonometry:
Trigonometry has a rich history dating back to ancient civilizations like the Babylonians, Egyptians, and Greeks. The Greek mathematician Hipparchus is often credited with the development of trigonometry. Over the centuries, trigonometry evolved with contributions from scholars like Aryabhata in India and Al-Khwarizmi in the Islamic world.
Types of Trigonometry:
There are various types of trigonometry, including:
- Plane Trigonometry: Deals with triangles on a flat plane.
- Spherical Trigonometry: Studies triangles on the surface of a sphere, important in navigation and astronomy.
- Analytical Trigonometry: Involves using algebraic and geometric techniques to solve trigonometric problems.
Trigonometric Formulas
1. Sine Formula: sin(θ) = opposite / hypotenuse
2. Cosine Formula: cos(θ) = adjacent / hypotenuse
3. Tangent Formula: tan(θ) = opposite / adjacent
4. Cosecant Formula: csc(θ) = 1 / sin(θ)
5. Secant Formula: sec(θ) = 1 / cos(θ)
6. Cotangent Formula: cot(θ) = 1 / tan(θ)
7. Pythagorean Identity: sin²(θ) + cos²(θ) = 1
8. Reciprocal Identity: sin(θ) = 1 / csc(θ), cos(θ) = 1 / sec(θ), tan(θ) = 1 / cot(θ)
9. Quotient Identity: tan(θ) = sin(θ) / cos(θ)
10. Co-Function Identity: sin(π/2 - θ) = cos(θ), cos(π/2 - θ) = sin(θ), tan(π/2 - θ) = cot(θ), cot(π/2 - θ) = tan(θ)
11. Even-Odd Identities: sin(-θ) = -sin(θ), cos(-θ) = cos(θ), tan(-θ) = -tan(θ)
12. Double Angle Identities: sin(2θ) = 2sin(θ)cos(θ), cos(2θ) = cos²(θ) - sin²(θ), tan(2θ) = 2tan(θ) / (1 - tan²(θ))
13. Half Angle Identities: sin(θ/2) = ±√((1 - cos(θ)) / 2), cos(θ/2) = ±√((1 + cos(θ)) / 2), tan(θ/2) = sin(θ) / (1 + cos(θ))
14. Trigonometric Addition Formulas: sin(A + B) = sin(A)cos(B) + cos(A)sin(B), cos(A + B) = cos(A)cos(B) - sin(A)sin(B)
15. Trigonometric Subtraction Formulas: sin(A - B) = sin(A)cos(B) - cos(A)sin(B), cos(A - B) = cos(A)cos(B) + sin(A)sin(B)
16. Product to Sum Formulas for Sine: sin(A)sin(B) = 1/2(cos(A - B) - cos(A + B))
17. Product to Sum Formulas for Cosine: cos(A)cos(B) = 1/2(cos(A - B) + cos(A + B))
18. Product to Sum Formulas for Sine and Cosine: sin(A)cos(B) = 1/2(sin(A + B) + sin(A - B))
19. Sum to Product Formulas for Sine: sin(A) + sin(B) = 2sin((A + B)/2)cos((A - B)/2)
20. Sum to Product Formulas for Cosine: cos(A) + cos(B) = 2cos((A + B)/2)cos((A - B)/2)
21. Trigonometric Equations: Solve equations like sin(x) = a, cos(x) = b, tan(x) = c, etc., for x.
22. Inverse Trigonometric Functions: sin⁻¹(x) = arcsin(x), cos⁻¹(x) = arccos(x), tan⁻¹(x) = arctan(x)
23. Derivatives of Trigonometric Functions: d/dx(sin(x)) = cos(x), d/dx(cos(x)) = -sin(x), d/dx(tan(x)) = sec²(x)
24. Integrals of Trigonometric Functions: ∫sin(x) dx = -cos(x) + C, ∫cos(x) dx = sin(x) + C, ∫tan(x) dx = -ln|cos(x)| + C
25. Law of Sines: a/sin(A) = b/sin(B) = c/sin(C)
26. Law of Cosines: c² = a² + b² - 2ab*cos(C)
27. Law of Tangents: (a - b)/(a + b) = (tan((A - B)/2))/(tan((A + B)/2))
28. Area of Triangle (Sine): A = (1/2)ab*sin(C)
29. Area of Triangle (Cosine): A = (1/2)ab*cos(C)
30. Area of Triangle (Tangent): A = (1/2)ab(tan(C/2))
31. Product of Sines Formula: sin(A)sin(B) = (1/2)(cos(A - B) - cos(A + B))
32. Product of Cosines Formula: cos(A)cos(B) = (1/2)(cos(A - B) + cos(A + B))
33. Product of Sine and Cosine Formula: sin(A)cos(B) = (1/2)(sin(A + B) + sin(A - B))
34. Triple Angle Formulas for Sine: sin(3θ) = 3sin(θ) - 4sin³(θ)
35. Triple Angle Formulas for Cosine: cos(3θ) = 4cos³(θ) - 3cos(θ)
36. Triple Angle Formulas for Tangent: tan(3θ) = (3tan(θ) - tan³(θ)) / (1 - 3tan²(θ))
37. Half Angle Formulas for Tangent: tan(θ/2) = ±√((1 - cos(θ)) / (1 + cos(θ)))
38. Double Angle Formulas for Secant: sec(2θ) = (1 + tan²(θ)) / (1 - tan²(θ))
39. Double Angle Formulas for Cosecant: csc(2θ) = (1 + cot²(θ)) / (1 - cot²(θ))
40. Double Angle Formulas for Cotangent: cot(2θ) = (cot²(θ) - 1) / (2cot(θ))
41. Sum and Difference Formulas for Tangent: tan(A ± B) = (tan(A) ± tan(B)) / (1 ∓ tan(A)tan(B))
42. Sum and Difference Formulas for Cotangent: cot(A ± B) = (cot(A)cot(B) ∓ 1) / (cot(B) ± cot(A))
43. Sum and Difference Formulas for Secant: sec(A ± B) = (sec(A)sec(B)) / (sec(B) ± sec(A))
44. Sum and Difference Formulas for Cosecant: csc(A ± B) = (csc(A)csc(B)) / (csc(A) ± csc(B))
45. Sum of Squares Formulas: sin²(θ) + cos²(θ) = 1, 1 + tan²(θ) = sec²(θ), 1 + cot²(θ) = csc²(θ)
46. Difference of Squares Formulas: sin²(θ) - cos²(θ) = -cos(2θ), tan²(θ) - 1 = -sec²(θ), cot²(θ) - 1 = -csc²(θ)
47. Half Angle Formulas for Secant: sec(θ/2) = ±√((1 + cos(θ)) / (1 - cos(θ)))
48. Half Angle Formulas for Cosecant: csc(θ/2) = ±√((1 - cos(θ)) / (1 + cos(θ)))
49. Half Angle Formulas for Cotangent: cot(θ/2) = ±√((1 + cos(θ)) / (1 - cos(θ)))
50. Sum and Difference of Secants: sec(A) + sec(B) = 2 / cos(A + B), sec(A) - sec(B) = 2sin(A - B) / (cos(A)cos(B))
51. Sum and Difference of Cosecants: csc(A) + csc(B) = 2sin(A + B) / (sin(A)sin(B)), csc(A) - csc(B) = 2sin(A - B) / (sin(A)sin(B))
52. Sum and Difference of Cotangents: cot(A) + cot(B) = (sin(A + B)) / (sin(A)sin(B)), cot(A) - cot(B) = (sin(A - B)) / (sin(A)sin(B))
53. Product-to-Sum Formulas for Tangent: tan(A)tan(B) = (tan(A) + tan(B)) / (1 - tan(A)tan(B)), tan(A)tan(B) = (tan(A) - tan(B)) / (1 + tan(A)tan(B))
54. Half Angle Formulas for Sine Squared: sin²(θ/2) = (1 - cos(θ)) / 2
55. Half Angle Formulas for Cosine Squared: cos²(θ/2) = (1 + cos(θ)) / 2
56. Half Angle Formulas for Tangent Squared: tan²(θ/2) = (1 - cos(θ)) / (1 + cos(θ))
57. Half Angle Formulas for Secant Squared: sec²(θ/2) = (1 + cos(θ)) / (1 - cos(θ))
58. Half Angle Formulas for Cosecant Squared: csc²(θ/2) = (1 - cos(θ)) / (1 + cos(θ))
59. Half Angle Formulas for Cotangent Squared: cot²(θ/2) = (cos(θ) - 1) / (cos(θ) + 1)
60. Versine Formulas: versin(θ) = 1 - cos(θ), versin(θ) = 2sin²(θ/2)
61. Haversine Formulas: haversin(θ) = sin²(θ/2)
62. Exsecant Formulas: exsec(θ) = sec(θ) - 1
63. Coversine Formulas: coversin(θ) = 1 - haversin(θ)
64. Law of Versines: a - 2bc*cos(A) = 2R*versin(A)
65. Law of Cotangents: cot(B) + cot(C) = (a + b + c) / (2A)
66. Symmetry Identities: sin(π - x) = sin(x), cos(π - x) = -cos(x), tan(π - x) = -tan(x), csc(π - x) = csc(x), sec(π - x) = -sec(x), cot(π - x) = -cot(x)
67. Periodicity Identities: sin(x + 2π) = sin(x), cos(x + 2π) = cos(x), tan(x + π) = tan(x), csc(x + 2π) = csc(x), sec(x + 2π) = sec(x), cot(x + π) = cot(x)
68. Angle Sum Formulas for Sine: sin(A + B) = sin(A)cos(B) + cos(A)sin(B), sin(A - B) = sin(A)cos(B) - cos(A)sin(B)
69. Angle Sum Formulas for Cosine: cos(A + B) = cos(A)cos(B) - sin(A)sin(B), cos(A - B) = cos(A)cos(B) + sin(A)sin(B)
70. Angle Difference Formulas for Tangent: tan(A - B) = (tan(A) - tan(B)) / (1 + tan(A)tan(B)), tan(A + B) = (tan(A) + tan(B)) / (1 - tan(A)tan(B))
71. Half Angle Formulas for Sine Squared: sin²(θ/2) = (1 - cos(θ)) / 2
72. Half Angle Formulas for Cosine Squared: cos²(θ/2) = (1 + cos(θ)) / 2
73. Half Angle Formulas for Tangent Squared: tan²(θ/2) = (1 - cos(θ)) / (1 + cos(θ))
74. Half Angle Formulas for Tangent: tan(θ/2) = ±√((1 - cos(θ)) / (1 + cos(θ)))
75. Half Angle Formulas for Cosine: cos(θ/2) = ±√((1 + cos(θ)) / 2)
76. Half Angle Formulas for Sine: sin(θ/2) = ±√((1 - cos(θ)) / 2)
77. Trigonometric Addition Formulas: sin(A + B) = sin(A)cos(B) + cos(A)sin(B), cos(A + B) = cos(A)cos(B) - sin(A)sin(B)
78. Trigonometric Subtraction Formulas: sin(A - B) = sin(A)cos(B) - cos(A)sin(B), cos(A - B) = cos(A)cos(B) + sin(A)sin(B)
79. Trigonometric Double Angle Formulas: sin(2θ) = 2sin(θ)cos(θ), cos(2θ) = cos²(θ) - sin²(θ), tan(2θ) = 2tan(θ) / (1 - tan²(θ))
80. Product to Sum Formulas for Sine: sin(A)sin(B) = (1/2)(cos(A - B) - cos(A + B))
81. Product to Sum Formulas for Cosine: cos(A)cos(B) = (1/2)(cos(A - B) + cos(A + B))
82. Product to Sum Formulas for Sine and Cosine: sin(A)cos(B) = (1/2)(sin(A + B) + sin(A - B))
83. Sum to Product Formulas for Sine: sin(A) + sin(B) = 2sin((A + B)/2)cos((A - B)/2)
84. Sum to Product Formulas for Cosine: cos(A) + cos(B) = 2cos((A + B)/2)cos((A - B)/2)
85. Sum to Product Formulas for Tangent: tan(A) + tan(B) = sin(A + B) / (cos(A)cos(B))
86. Difference to Product Formulas for Sine: sin(A) - sin(B) = 2cos((A + B)/2)sin((A - B)/2)
87. Difference to Product Formulas for Cosine: cos(A) - cos(B) = -2sin((A + B)/2)sin((A - B)/2)
88. Half Angle Formulas for Tangent: tan(θ/2) = ±√((1 - cos(θ)) / (1 + cos(θ)))
89. Half Angle Formulas for Secant: sec(θ/2) = ±√((1 + cos(θ)) / (1 - cos(θ)))
90. Half Angle Formulas for Cosecant: csc(θ/2) = ±√((1 - cos(θ)) / (1 + cos(θ)))
91. Double Angle Formulas for Tangent: tan(2θ) = 2tan(θ) / (1 - tan²(θ))
92. Double Angle Formulas for Secant: sec(2θ) = (1 + tan²(θ)) / (1 - tan²(θ))
93. Double Angle Formulas for Cosecant: csc(2θ) = (1 + cot²(θ)) / (1 - cot²(θ))
94. Triple Angle Formulas for Sine: sin(3θ) = 3sin(θ) - 4sin³(θ)
95. Triple Angle Formulas for Cosine: cos(3θ) = 4cos³(θ) - 3cos(θ)
96. Triple Angle Formulas for Tangent: tan(3θ) = (3tan(θ) - tan³(θ)) / (1 - 3tan²(θ))
97. Addition Formulas for Tangent: tan(A + B) = (tan(A) + tan(B)) / (1 - tan(A)tan(B))
98. Subtraction Formulas for Tangent: tan(A - B) = (tan(A) - tan(B)) / (1 + tan(A)tan(B))
99. Power Reduction Formulas for Sine and Cosine: cos²(θ) = (1 + cos(2θ)) / 2, sin²(θ) = (1 - cos(2θ)) / 2
100. Power Reduction Formulas for Tangent: tan²(θ) = (1 - cos(2θ)) / (1 + cos(2θ))
101. Angle Sum Formulas for Tangent: tan(A + B) = (tan(A) + tan(B)) / (1 - tan(A)tan(B)), tan(A - B) = (tan(A) - tan(B)) / (1 + tan(A)tan(B))
102. Angle Sum Formulas for Secant: sec(A + B) = (sec(A)sec(B)) / (sec(B) ± sec(A)), sec(A - B) = (sec(A)sec(B)) / (sec(B) ± sec(A))
103. Angle Sum Formulas for Cosecant: csc(A + B) = (csc(A)csc(B)) / (csc(A) ± csc(B)), csc(A - B) = (csc(A)csc(B)) / (csc(A) ± csc(B))
104. Power Reduction Formulas for Sine and Cosine: sin²(θ) = (1 - cos(2θ)) / 2, cos²(θ) = (1 + cos(2θ)) / 2
105. Sum and Difference of Cosines: cos(A) + cos(B) = 2cos((A + B)/2)cos((A - B)/2), cos(A) - cos(B) = -2sin((A + B)/2)sin((A - B)/2)
106. Sum and Difference of Sines: sin(A) + sin(B) = 2sin((A + B)/2)cos((A - B)/2), sin(A) - sin(B) = 2cos((A + B)/2)sin((A - B)/2)
107. Angle Difference Formulas for Secant and Cosecant: sec(A - B) = (sec(A)sec(B)) / (sec(A) ± sec(B)), csc(A - B) = (csc(A)csc(B)) / (csc(A) ± csc(B))
108. Half Angle Formulas for Tangent: tan(θ/2) = ±√((1 - cos(θ)) / (1 + cos(θ)))
109. Half Angle Formulas for Secant: sec(θ/2) = ±√((1 + cos(θ)) / (1 - cos(θ)))
110. Half Angle Formulas for Cosecant: csc(θ/2) = ±√((1 - cos(θ)) / (1 + cos(θ)))
111. Half Angle Formulas for Cosine: cos(θ/2) = ±√((1 + cos(θ)) / 2)
112. Half Angle Formulas for Sine: sin(θ/2) = ±√((1 - cos(θ)) / 2)
113. Half Angle Formulas for Cotangent: cot(θ/2) = ±√((1 + cos(θ)) / (1 - cos(θ)))
114. Double Angle Formulas for Tangent: tan(2θ) = 2tan(θ) / (1 - tan²(θ))
115. Double Angle Formulas for Secant: sec(2θ) = (1 + tan²(θ)) / (1 - tan²(θ))
116. Double Angle Formulas for Cosecant: csc(2θ) = (1 + cot²(θ)) / (1 - cot²(θ))
117. Triple Angle Formulas for Sine: sin(3θ) = 3sin(θ) - 4sin³(θ)
118. Triple Angle Formulas for Cosine: cos(3θ) = 4cos³(θ) - 3cos(θ)
119. Triple Angle Formulas for Tangent: tan(3θ) = (3tan(θ) - tan³(θ)) / (1 - 3tan²(θ))
120. Sum and Difference of Tangents: tan(A + B) = (tan(A) + tan(B)) / (1 - tan(A)tan(B)), tan(A - B) = (tan(A) - tan(B)) / (1 + tan(A)tan(B))
121. Sum and Difference of Secants: sec(A + B) = (sec(A)sec(B)) / (sec(B) ± sec(A)), sec(A - B) = (sec(A)sec(B)) / (sec(B) ± sec(A))
122. Sum and Difference of Cosecants: csc(A + B) = (csc(A)csc(B)) / (csc(A) ± csc(B)), csc(A - B) = (csc(A)csc(B)) / (csc(A) ± csc(B))
123. Sum and Difference of Cotangents: cot(A + B) = (cot(A)cot(B) - 1) / (cot(B) ± cot(A)), cot(A - B) = (cot(A)cot(B) + 1) / (cot(B) ± cot(A))
124. Pythagorean Identity: sin²(θ) + cos²(θ) = 1
125. Cofunction Identities: sin(π/2 - θ) = cos(θ), cos(π/2 - θ) = sin(θ), tan(π/2 - θ) = cot(θ)
126. Angle Sum and Difference for Tangent: tan(A + B) = (tan(A) + tan(B)) / (1 - tan(A)tan(B)), tan(A - B) = (tan(A) - tan(B)) / (1 + tan(A)tan(B))
127. Half Angle Formulas for Secant and Cosecant: sec(θ/2) = ±√((1 + cos(θ)) / (1 - cos(θ))), csc(θ/2) = ±√((1 - cos(θ)) / (1 + cos(θ)))
128. Angle Sum and Difference for Secant and Cosecant: sec(A + B) = (sec(A)sec(B)) / (sec(A) ± sec(B)), sec(A - B) = (sec(A)sec(B)) / (sec(A) ± sec(B))
129. Angle Sum and Difference for Cotangent: cot(A + B) = (cot(A)cot(B) - 1) / (cot(B) ± cot(A)), cot(A - B) = (cot(A)cot(B) + 1) / (cot(B) ± cot(A))
130. Pythagorean Identity for Tangent: 1 + tan²(θ) = sec²(θ)
131. Pythagorean Identity for Cotangent: 1 + cot²(θ) = csc²(θ)
132. Reciprocal Identities: sec(θ) = 1 / cos(θ), csc(θ) = 1 / sin(θ), cot(θ) = 1 / tan(θ)
133. Double Angle Formulas for Sine and Cosine: sin(2θ) = 2sin(θ)cos(θ), cos(2θ) = cos²(θ) - sin²(θ)
134. Double Angle Formulas for Tangent: tan(2θ) = 2tan(θ) / (1 - tan²(θ))
135. Half Angle Formulas for Tangent: tan(θ/2) = ±√((1 - cos(θ)) / (1 + cos(θ)))