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NCERT Solutions for Class 12 Mathematics Chapter 2 Inverse Trigonometric Functions is based on basic concepts of inverse trigonometric functions, properties of inverse trigonometric functions, and miscellaneous examples. Inverse trigonometric functions are defined as inverse functions of the basic trigonometric functions which are sine, cosine, tangent, cotangent, secant, and cosecant functions. They are also termed arcus functions, antitrigonometric functions, or cyclometric functions.
- NCERT Solutions for Class 12 Mathematics Chapter 2 Inverse Trigonometric Functions will carry a weightage of around 4-8 marks in the CBSE Class 12 Examination.
- Short answers and MCQ questions can come from range, domain, principal value branch, graphs of inverse trigonometric functions, and elementary properties of inverse trigonometric functions.
Download PDF: NCERT Solutions for Class 12 Mathematics Chapter 2
NCERT Solutions Class 12 Mathematics Chapter 2 Important Topics
Inverse trigonometry is an important topic in the NCERT syllabus for Class 12 students. Solutions to Class 12 Maths Chapter 2 are prepared as per the NCERT curriculum. These solutions are carefully prepared in such a way that it provides students with a step-by-step approach to solving any problems. The topic-wise explanation has been shared below:
Circular Representation of Inverse Trigonometric Functions
Convention symbol represents inverse trigonometric function using arc-prefixes such as: arcsin(x), arccos(x), arctan(x), arccsc(x), arcsec(x), arccot(x). sin-1x, cos-1x, tan-1x etc. These are also called as arcsin x, arccosine x etc.
Here are some notable properties and vice-versa properties:
| Properties | Vice-versa Properties |
|---|---|
| If sin θ=x (-1≤x≤1) then θ=sin-1(x) | sin-1(x)=θ then sin θ=x |
| If cos θ=x (-1≤x≤1) then θ=cos-1(x) | cos-1(x)= θ then cos θ=x |
| If tan θ=x (-∞<x </x) then θ=tan-1(x) | tan-1(x)= θ then tan θ=x |
| If cot θ=x (-∞<x </x) then θ=cot-1(x) | cot-1(x)= θ then cot θ=x |
| If cosec θ=x (|x|≥ 1) then θ=cosec-1(x) | cosec-1(x)= θ then cosec θ=x |
| If sec θ=x (|x|≥ 1) then θ=sec-1(x) | sec-1(x)= θ then sec θ=x |
Trigonometric functions are used for finding out measurements like heights of tall buildings, etc. without using measurement tools. Similarly, inverse functions are widely used in Engineering and other sciences including Physics. For example, the oscilloscope is an electronic device that can convert electrical signals into graphs like that of sine function. By manipulating controls, amplitude can be changed along with period and phase shift of sine curves.
Here are some important notes from the topic:
| Inverse sine function | Inverse cosine function | Inverse Tangent function | Inverse Cosecant function | Inverse secant function | Inverse Cot function |
|---|---|---|---|---|---|
| Domain: [-1,1] | Domain: [-1,1] | Domain: R | Domain: (-∞,-1] U [1,∞) | Domain: (-∞,-1] U [1,∞) | Domain: R |
| Range: [-π/2, π/2] | Range: [0,π] | Range: [-π/2, π/2] | Range: [-π/2, π/2]-{0} | Range: [0, π]-{π/2} | Range: (0, π) |
Derivative of Inverse Trigonometric Functions
Inverse Trigonometric functions can be differentiated and the derivatives can be found. Here are some common derivatives:
Inverse Trigonometric Functions
Inverse Trigonometric Functions include sine, cosine, tangent, cotangent, secant and cosecant. These functions are also called arcus functions, anti trigonometric functions or cyclometer functions. Inverse trigonometry can be applied across various fields including physics, geometry, engineering, navigation, aviation, marine biology etc. Inverse trigonometry can help with obtaining angles of a triangle from any trigonometric function.
Basic formulas of Inverse Trigonometric Functions are:
- sin-1(-x) = -sin-1(x), x ∈ [-1, 1]
- tan-1(-x) = -tan-1(x), x ∈ R
- cos-1(-x) = π -cos-1(x), x ∈ [-1, 1]
- cot-1(-x) = π – cot-1(x), x ∈ R
- sec-1(-x) = π -sec-1(x), |x| ≥ 1
- cosec-1(-x) = -cosec-1(x), |x| ≥ 1
Half-Angle Formula
Half-angle formulas are derived from double angle formulas. There are a wide variety of double angle formulas in the form of 2θ, 2A, 2x, etc. Double angle formulas for sin, cos, and tan are:
- Sin 2x = 2 sin x cos x
- Cos 2x = cos2 x - sin2 x (or)
= 1 - 2 sin2x (or)
= 2 cos2x - 1
Inverse Tan
Inverse Tan or Inverse Tangent is the inverse function of trigonometric function Tangent. Inverse tangent is denoted by tan-1 (x). Tangent is the ratio of the perpendicular and base of a triangle. All inverse trigonometric functions are known as arcus functions. Thus, tan-1 (x) can also be written as arctan. These functions can be applicable in construction, engineering, architecture, cartography etc.
Check out the graph of Inverse Tangent Function:
Graph of Inverse Tangent
NCERT Solutions For Class 12 Maths Chapter 2 Exercises
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