Inverse Trigonometric Formulas

About Inverse Trigonometric Formulas

Trigonometry is a branch of mathematics that teaches us about the relationships between the angles and sides of a right-angled triangle. You'll find a collection of trigonometry formulas based on functions and ratios like sin, cos, and tan in the Maths syllabus for Class 11 and 12. Inverse trigonometry principles have also been taught to us. Sin-1x, cos-1x, cot-1 x, tan-1 x, cosec-1 x, sec-1 x are the inverse trigonometric functions.

What do you mean by Inverse Trigonometric Function?

  • The anti trigonometric functions, also known as arcus functions or cyclometric functions, are inverse trigonometric functions. To get the angle of a triangle using any of the trigonometric functions, utilise the inverse trigonometric functions sine, cosine, tangent, cosecant, secant, and cotangent. It is frequently utilised in a variety of domains, including geometry, engineering, and physics. However, most of the time, the inverse trigonometric function is represented by an arc-prefix symbol such as arcsin(x), arccos(x), arctan(x), arccsc(x), arcsec(x), arccot(x).
  • Consider, the function as y = f(x), and x = g(y) then the inverse function is written as g = f-1,
  • This means that if y = f(x), then x = f-1(y).
  • Such that f(g(y)) = y and g(f(y)) = x.
  • Example of Inverse trigonometric functions: x = sin-1y
The list of inverse trigonometric functions with domain and range value is given below:
Functions Domain Range
Sin-1x [-1, 1] [-π/2, π/2]
Cos-1x [-1, 1] [0, π/2]
Tan-1 x R [-π/2, π/2]
Cosec-1 x R - (-1,1) [-π/2, π/2]
Sec-1 R - (-1,1) [0,π]-{ π/2}
Cot-1 x R [-π/2, π/2]-{0}

List of Inverse Trigonometric Formulas

Inverse trigonometry formulae are constructed from the basic properties of trigonometry to solve various forms of inverse trigonometric functions. Below is a list of formulas to help you answer the problems.
S. NO. Inverse Trigonometric Formulas
1. sin-1(-x) = -sin-1(x), x ∈ [-1, 1]
2. cos-1(-x) = π -cos-1(x), x ∈ [-1, 1]
3. tan-1(-x) = -tan-1(x), x ∈ R
4. cosec-1(-x) = -cosec-1(x), |x| ≥ 1
5. sec-1(-x) = π -sec-1(x), |x| ≥ 1
6. cot-1(-x) = π – cot-1(x), x ∈ R
7. sin-1x + cos-1x = π/2 , x ∈ [-1, 1]
8. tan-1x + cot-1x = π/2 , x ∈ R
9. sec-1x + cosec-1x = π/2 ,|x| ≥ 1
10. sin-1(1/x) = cosec-1(x), if x ≥ 1 or x ≤ -1
11. cos-1(1/x) = sec-1(x), if x ≥ 1 or x ≤ -1
12. tan-1(1/x) = cot-1(x), x > 0
13. tan-1x + tan-1y = tan-1((x+y)/(1-xy)), if the value xy < 1
14. tan-1 x – tan-1 y = tan-1((x-y)/(1+xy)), if the value xy > -1
15. 2 tan-1 x = sin-1(2x/(1+x2)), |x| ≤ 1
16. 2tan-1 x = cos-1((1-x2)/(1+x2)), x ≥ 0
17. 2tan-1 x = tan-1(2x/(1-x2)), -1 < x < 1
18. 3sin-1x = sin-1(3x-4x3)
19. 3cos-1x = cos-1(4x3-3x)
20. 3tan-1x = tan-1((3x-x3)/(1-3x2))
21. sin(sin-1(x)) = x,-1 ≤ x ≤ 1
22. cos(cos-1(x)) = x, -1 ≤ x ≤ 1
23. tan(tan-1(x)) = x, – ∞ < x < ∞.
24. cosec(cosec-1(x)) = x, – ∞ < x ≤ 1 or -1 ≤ x < ∞
25. sec(sec-1(x)) = x,- ∞ < x ≤ 1 or 1 ≤ x < ∞
26. cot(cot-1(x)) = x, – ∞ < x < ∞
27. sin-1(sin θ) = θ, -π/2 ≤ θ ≤ π/2
28. cos-1(cos θ) = θ, 0 ≤ θ ≤ π
29. tan-1(tan θ) = θ, -π/2 < θ < π/2
30. cosec-1(cosec θ) = θ, – π/2 ≤ θ < 0 or 0 < θ ≤ π/2
31. sec-1(sec θ) = θ, 0 ≤ θ ≤ π/2 or π/2 < θ ≤ π
32. cot-1(cot θ) = θ, 0 < θ < π
33. sin-1x + sin-1y = sin-1(x √1 - y2 + y √1 - x2), if x,y ≥ 0 and x2 + y2 ≤ 1
34. sin-1x + sin-1y = π - sin-1(x √1 - y2 + y √1 - x2), if x,y ≥ 0 and x2 + y2 > 1
35. sin-1x - sin-1y = π - sin-1(x √1 - y2 - y √1 - x2), if x,y ≥ 0 and x2 + y2 ≤ 1
36. sin-1x - sin-1y = π - sin-1(x √1 - y2 - y √1 - x2), if x,y ≥ 0 and x2 + y2 > 1
37. cos-1x + cos-1y = cos-1(xy - √1 - x2 √1 - y2), if y > 0 and x2 + y2 ≤ 1
38. cos-1x + cos-1y = π - cos-1(xy - √1 - x2 √1 - y2), if x,y > 0 and x2 + y2 > 1
39. cos-1x - cos-1y = cos-1(xy + √1 - x2 √1 - y2), if x,y > 0 and x2 + y2 ≤ 1
40. cos-1x + cos-1y = π - cos-1(xy + √1 - x2 √1 - y2), if x,y > 0 and x2 + y2 > 1

 

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