Inverse Trigonometric Functions
Inverse Trigonometric Functions of NCERT Class 12 Maths, gives an account of various topics such as the remarks based on basic concepts of inverse trigonometric functions, properties of inverse trigonometric functions, and miscellaneous examples. Inverse Trigonometric Functions are quite useful in Calculus to define different integrals. You can also check the Trigonometric Formulas here.
Inverse trigonometric functions: Inverse trigonometric functions map real numbers back to angles. e.g. Inverse of sine function denoted by sin-1 or arc sin(x) is defined on [-1,1].
Some of the important useful properties of Inverse Trigonometric Functions are:
Functions | Domain | Range |
---|---|---|
y = sin-1 x | [–1, 1] | [−π/2,π/2] |
y = cos-1 x | [–1, 1] | [0,π] |
y = cosec-1 x | R – (–1, 1) | [−π/2, π/2] – {0} |
y = sec-1 x | R – (–1, 1) | [0,π] – {π/2, π/2} |
y = tan-1 x | R | (−π/2, π/2) |
y = cot-1 x | R | (0,π) |
Self-Adjusting Trigonometric Property
Self Adjusting Inverse Trigonometric Properties are,
- sin(sin-1 x)=x
- sin-1 (sin x) = x
- cos(cos-1 x) = x
- cos-1 (cos x) = x
- tan(tan-1 x) = x
- tan-1 (tan x) = x
- sec(sec-1x) = x
- sec-1(sec x) = x
- cosec-1(cosec x) = x
- cosec(cosec-1 x) = x
- cot-1(cot x) = x
- cot(cot-1x) = x
Reciprocal Relations
Reciprocal Relations of the Inverse Trigonometric Relations are,
- sin-1 (1/x) = cosec-1 x, x ≥ 1 or x ≤ -1
- cos-1 (1/x) = sec-1 x, x ≥ 1 or x ≤ -1
- tan-1 (1/x) = cot-1 x, x > 0
Even and Odd Functions
Even and Odd Functions of the Inverse Trigonometry Functions are,
- sin-1 (-x) = -sin-1 (x), x ∈ [-1, 1]
- tan-1 (-x) = -tan-1 (x), x ∈ R
- cosec-1 (-x) = -cosec-1 (x), |x| ≥1
- cos-1 (-x) = π – cos-1 (x), x ∈ [-1, 1]
- sec-1 (-x) = π – sec-1 (x), |x| ≥1
- cot-1 (-x) = π – cot-1 (x), x ∈ R
Complementary Relations
The complementary relation of the Inverse trigonometry functions is,
- sin-1 x + cos-1 x = π/2
- tan-1 x + cot-1 x = π/2
- cosec-1 x + sec-1 x = π/2
Sum and Difference Formulae
The sum and difference formulas are the important formulas used in inverse trigonometric functions, some of the important inverse trigonometric sum and difference formulas are,
- tan-1 x + tan-1 y = tan-1 {(x+y)/(1−xy)}
- tan-1 x – tan-1 y = tan-1 {(x-y)/(1+xy)}
- sin-1 x + sin-1 y = sin-1 [x√(1-y2)+y√(1-x2)]
- sin-1 x – sin-1 y = sin-1 [x√(1-y2)-y√(1-x2)]
- cos-1 x + cos-1 y = cos-1 [xy-√(1-x2)√(1-y2)]
- cos-1 x – cos-1 y = cos-1 [xy+√(1-x2)√(1-y2)]
- cot-1 x + cot-1 y = cot-1 [(xy-1)/(x+y)]
- cot-1 x + cot-1 y = cot-1 [(xy+1)/(y-x)]
Double Angle Formula
The double angle formula for the inverse trigonometric functions is,
- 2tan-1 x = sin-1 (2x/1+x2)
- 2tan-1 x = cos-1 (1-x2/1+x2)
- 2tan-1 x = tan-1 (2x/1-x2)
- 2sin-1 x = sin-1 (2x√(1+x2))
- 2cos-1 x = sin-1 (2x√(1-x2))
Conversion Properties
The conversion properties for the inverse trigonometric function are,
- sin-1 x = cos-1 √(1-x2) = tan-1 {x/x√(1-x2)} = cot-1 {√(1-x2)/x}
- cos-1 x = sin-1 √(1-x2) = tan-1 {√(1-x2)/x} = cot-1 {x/√(1-x2)}
- tan-1 x = sin-1 {x/√(1-x2)} = cos-1 {x/√(1+x2)} = sec-1 √(1+x2) = cosec-1 {√(1+x2)/x}
Class 12 Maths Formulas
Class 12 maths formulas page is designed for the convenience of the learners so that one can understand all the important concepts of Class 12 Mathematics directly and easily. Math formulae for Class 12 are for the students who find mathematics to be a nightmare and difficult to grasp. They may become hesitant and lose interest in studies as a result of this. We have included all of the key formulae for the 12th standard Maths topic, which students may simply recall, to assist them in understanding Maths in a straightforward manner. For all courses such as Integration, Differentiation, Trignometry, Relation and Functions, and so on, the formulae are provided here according to the NCERT curriculum.
Table of Content
- Chapter 1: Relations and Functions
- Chapter 2: Inverse Trigonometric Functions
- Chapter 3: Matrices
- Chapter 4: Determinants
- Chapter 5: Continuity and Differentiability
- Chapter 6: Applications of Derivatives
- Chapter 7: Integrals
- Chapter 8: Applications of Integrals
- Chapter 9: Differential Equations
- Chapter 10: Vector Algebra
- Chapter 11: Three-Dimensional Geometry
- Chapter 12: Linear Programming
- Chapter 13: Probability
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