All Trigonometry Formula

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Trigonometry Formula 

 Here are some of the fundamental trigonometric formulas:


1. Pythagorean identities:

  • sin²θ + cos²θ = 1

 • tan²θ + 1 = sec²θ

 •  1 + cot²θ = csc²θ


2. Angle sum and difference formulas:

 •  sin(A + B) = sin A * cos B + cos A * sin B

 •  sin(A - B) = sin A * cos B - cos A * sin B

 •  cos(A + B) = cos A * cos B - sin A * sin B

 •  cos(A - B) = cos A * cos B + sin A * sin B

 •  tan(A + B) = (tan A + tan B) / (1 - tan A * tan B)

•   tan(A - B) = (tan A - tan B) / (1 + tan A * tan B)


3. Double angle formulas:

  • sin(2θ) = 2 * sin θ * cos θ

• cos(2θ) = cos²θ - sin²θ = 2 * cos²θ - 1 = 1 - 2 * sin²θ

 • tan(2θ) = (2 * tan θ) / (1 - tan²θ)


4. Half-angle formulas:

   • sin(θ/2) = ±√[(1 - cos θ) / 2]

   • cos(θ/2) = ±√[(1 + cos θ) / 2]

   •tan(θ/2) = ±√[(1 - cos θ) / (1 + cos θ)]


5. Law of Sines:

  • sin A / a = sin B / b = sin C / c

   (Here, A, B, C are angles, and a, b, c are the corresponding side lengths of a triangle.)


6. Law of Cosines:

  • c² = a² + b² - 2ab * cos C

   (Here, c is the side opposite angle C, and a, b are the other two sides.)


7. Sum-to-Product and Product-to-Sum formulas:

 •  sin A + sin B = 2 * sin[(A + B) / 2] * cos[(A - B) / 2]

 •  sin A - sin B = 2 * cos[(A + B) / 2] * sin[(A - B) / 2]

 •  cos A + cos B = 2 * cos[(A + B) / 2] * cos[(A - B) / 2]

 •   cos A - cos B = -2 * sin[(A + B) / 2] * sin[(A - B) / 2]


8. Cofunction identities:

   • sin(π/2 - θ) = cos θ

   • cos(π/2 - θ) = sin θ

   • tan(π/2 - θ) = 1 / tan θ

   • sec(π/2 - θ) = 1 / cos θ

   • csc(π/2 - θ) = 1 / sin θ

   • cot(π/2 - θ) = 1 / tan θ


9. Periodicity identities:

   • sin (θ + 2π) = sin θ

   • cos (θ + 2π) = cos θ

   • tan (θ + π) = tan θ


10. Reciprocal identities:

    • csc θ = 1 / sin θ

    • sec θ = 1 / cos θ

    • cot θ = 1 / tan θ


11. Even-Odd identities:

    • sin(-θ) = -sin θ

    • cos(-θ) = cos θ

    • tan(-θ) = -tan θ


12. Law of Tangents:

    • (a - b) / (a + b) = tan[(A - B) / 2]

    • (a + b) / (a - b) = cot[(A - B) / 2]

    (Here, a, b are side lengths opposite angles A and B, respectively.)


13. Law of Cotangents:

    • (a + b) / (a - b) = tan[(π/2 - A - B) / 2]

    • (a - b) / (a + b) = cot[(π/2 - A - B) / 2]

    (Here, a, b are side lengths opposite angles A and B, respectively.)


14. Euler's Formula:

    • e^(ix) = cos x + i * sin x

    (Here, e is the base of the natural logarithm, i is the imaginary unit, and x is an angle in radians.)


15. Law of Sines for Spherical Trigonometry:

    • sin(a) / sin(A) = sin(b) / sin(B) = sin(c) / sin(C)

    (Here, a, b, c are sides of a spherical triangle, and A, B, C are the opposite angles.)


16. Law of Cosines for Spherical Trigonometry:

    • cos(c) = cos(a) * cos(b) + sin(a) * sin(b) * cos(C)

    (Here, a, b, c are sides of a spherical triangle, and C is the included angle.)


17. Hyperbolic Identities:

    • sinh x = (e^x - e^(-x)) / 2

    • cosh x = (e^x + e^(-x)) / 2

    • tanh x = sinh x / cosh x

    (Here, sinh represents hyperbolic sine, cosh represents hyperbolic cosine, and tanh represents hyperbolic tangent.)


18. Addition and Subtraction Formulas:

    • sin(A + B) = sin A * cos B + cos A * sin B

    • sin(A - B) = sin A * cos B - cos A * sin B

    • cos(A + B) = cos A * cos B - sin A * sin B

    • cos(A - B) = cos A * cos B + sin A * sin B

    • tan(A + B) = (tan A + tan B) / (1 - tan A * tan B)

    • tan(A - B) = (tan A - tan B) / (1 + tan A * tan B)


19. Product-to-Sum and Sum-to-Product Formulas:

    • sin A * sin B = (1/2) * [cos(A - B) - cos(A + B)]

    • cos A * cos B = (1/2) * [cos(A - B) + cos(A + B)]

    • sin A * cos B = (1/2) * [sin(A + B) + sin(A - B)]


20. Law of Cosines:

    • c^2 = a^2 + b^2 - 2ab * cos C

    (Here, a, b, and c are the side lengths of a triangle, and C is the angle opposite side c.)


21. Law of Sines:

    • a / sin A = b / sin B = c / sin C

    (Here, a, b, and c are the side lengths of a triangle, and A, B, and C are the opposite angles.)


22. Heron's Formula for the Area of a Triangle:

    • Area = √[s * (s - a) * (s - b) * (s - c)]

    (Here, s is the semi-perimeter of the triangle, and a, b, and c are the side lengths.)


23. Law of Tangents:

    • (a - b) / (a + b) = tan[(A - B) / 2]

    (Here, a and b are side lengths opposite angles A and B, respectively.)


24. Law of Cotangents:

    • (a + b) / (a - b) = tan[(π/2 - A - B) / 2]

    (Here, a and b are side lengths opposite angles A and B, respectively.)


25. Law of Cosines for Spherical Trigonometry:

    • cos(c) = cos(a) * cos(b) + sin(a) * sin(b) * cos(C)

    (Here, a, b, and c are sides of a spherical triangle, and C is the included angle.)


26. Law of Haversines for Spherical Trigonometry:

    • haversin(c) = haversin(a - b) + sin(a) * sin(b) * haversin(C)

    (Here, a, b, and c are sides of a spherical triangle, and C is the included angle.)


27. Law of Versines for Spherical Trigonometry:

 •  versin(c) = versin(a) + versin(b) - 2 * sin(a) * sin(b) * versin(C / 2)²

    (Here, a, b, and c are sides of a spherical triangle, and C is the included angle.)


28. Triple Angle Formulas:

    • sin(3θ) = 3 * sin θ - 4 * sin³θ

    • cos(3θ) = 4 * cos³θ - 3 * cos θ

    • tan(3θ) = (3 * tan θ - tan³θ) / (1 - 3 * tan²θ)


29. Inverse Trigonometric Function Formulas:

    • arcsin x + arccos x = π/2

    • arctan x + arccot x = π/2

    •  arcsin x = arccos √(1 - x²)

    • arccos x = arcsin √(1 - x²)

    • arctan x = arccot(1/x)


30. Half-Angle Formulas:

    • sin(θ/2) = ±√[(1 - cos θ) / 2]

    • cos(θ/2) = ±√[(1 + cos θ) / 2]

    • tan(θ/2) = ±√[(1 - cos θ) / (1 + cos θ)]


31. Cofunction Identities:

    • sin(π/2 - θ) = cos θ

    • cos(π/2 - θ) = sin θ

    • tan(π/2 - θ) = 1 / tan θ

    • sec(π/2 - θ) = 1 / cos θ

    • csc(π/2 - θ) = 1 / sin θ

    • cot(π/2 - θ) = 1 / tan θ


32. Product of Sines Formula:

    • sin(A) * sin(B) = (1/2) * [cos(A - B) - cos(A + B)]


33. Product of Cosines Formula:

    • cos(A) * cos(B) = (1/2) * [cos(A - B) + cos(A + B)]


34. Triple Product Formula:

    • cos(A) * cos(B) * cos(C) = (1/2) * [cos(A + B + C) + cos(A - B + C) + cos(A + B - C) + cos(A - B - C)]


35. De Moivre's Formula:

 •  (cos θ + i * sin θ)^n = cos(nθ) + i * sin(nθ)

    (Here, i is the imaginary unit and n is a real number.)


36. Exponential Formulas:

    • sin x = (e^(ix) - e^(-ix)) / (2i)

    • cos x = (e^(ix) + e^(-ix)) / 2

    • tan x = (e^(ix) - e^(-ix)) / (i * (e^(ix) + e^(-ix)))

    (Here, e is the base of the natural logarithm and i is the imaginary unit.)


37. 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

    • tan(A + B) = (tan A + tan B) / (1 - tan A * tan B)


38. Trigonometric Double Angle Formulas:

    • sin(2θ) = 2 * sin θ * cos θ

    • cos(2θ) = cos²θ - sin²θ = 2 * cos²θ - 1 = 1 - 2 * sin²θ

    • tan(2θ) = (2 * tan θ) / (1 - tan²θ)


39. Trigonometric Half Angle Formulas:

    • sin(θ/2) = ±√[(1 - cos θ) / 2]

    • cos(θ/2) = ±√[(1 + cos θ) / 2]

    • tan(θ/2) = ±√[(1 - cos θ) / (1 + cos θ)]


40. Law of Sines for Spherical Trigonometry:

    • sin(a) / sin(A) = sin(b) / sin(B) = sin(c) / sin(C)

    (Here, a, b, and c are sides of a spherical triangle, and A, B, and C are the opposite angles.)


41. Law of Cosines for Spherical Trigonometry:

    • cos(c) = cos(a) * cos(b) + sin(a) * sin(b) * cos(C)

    (Here, a, b, and c are sides of a spherical triangle, and C is the included angle.)


42. Law of Versed Sines for Spherical Trigonometry:

    • versin(c) = 1 - cos(c)

    • versin(c) = 2 * sin²(c/2)

    (Here, c is the angle in radians.)



43. Power Reduction Formulas:

    • cos²θ = (1 + cos(2θ)) / 2

    • sin²θ = (1 - cos(2θ)) / 2

    • tan²θ = (1 - cos(2θ)) / (1 + cos(2θ))


44. Product-to-Sum Formulas:

    • sin A * sin B = (1/2) * [cos(A - B) - cos(A + B)]

    • cos A * cos B = (1/2) * [cos(A - B) + cos(A + B)]

    • sin A * cos B = (1/2) * [sin(A + B) + sin(A - B)]


45. Sum-to-Product Formulas:

    • sin A + sin B = 2 * sin[(A + B) / 2] * cos[(A - B) / 2]

    • sin A - sin B = 2 * cos[(A + B) / 2] * sin[(A - B) / 2]

    • cos A + cos B = 2 * cos[(A + B) / 2] * cos[(A - B) / 2]

    • cos A - cos B = -2 * sin[(A + B) / 2] * sin[(A - B) / 2]


46. Law of Tangents:

    • (a - b) / (a + b) = tan[(A - B) / 2]

    (Here, a and b are side lengths opposite angles A and B, respectively.)


47. Law of Cotangents:

    • (a + b) / (a - b) = tan[(π/2 - A - B) / 2]

    (Here, a and b are side lengths opposite angles A and B, respectively.)


48. Law of Exponents:

    • sin^n θ = (sin θ)^n

    • cos^n θ = (cos θ)^n

    • tan^n θ = (tan θ)^n

    (Here, n is a positive integer.)


These additional formulas cover power reduction formulas, product-to-sum and sum-to-product formulas, the law of tangents, the law of cotangents, and the law of exponents. Trigonometry encompasses a wide range of relationships and identities, and these formulas provide additional tools for solving trigonometric problems. If you have any specific questions or would like further explanations, please let me know!

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