NCERT Solutions for Class 12 Maths Chapter 2 – Inverse Trigonometric Functions

October 17, 2020 by Jahnavi

Category: Class 12, NCERT Solutions 0 Comments

Chapter 2 of NCERT Class 12 Maths Inverse Trigonometric Functions plays an important role in calculus. Students must practice calculus part of class 12 to score excellent marks in the board exams. NCERT solutions provide solutions for all the problems from chapter 2 of class 12 inverse trigonometric functions. Students can download NCERT solutions PDF of class 12 inverse trigonometric functions, and practice for themselves at any instant of time. The concepts of Inverse trigonometry functions are also used in the higher classes in the streams of science and engineering.

Practicing calculus is very useful to score more marks because the calculus part has more weightage in the board exams as well as many competitive exams. For most of the competitive exams, the questions are given from the topics picked from the NCERT books. NCERT solutions are very clear and easy to understand for the students. Any problem from the Inverse Trigonometric Functions of NCERT Class 12 Maths can be checked from the PDF below if you got stuck anywhere in between solving problems.

NCERT Solutions for Class 12 Maths Chapter 2 – Inverse Trigonometric Functions

Topics covered: chapter 2 inverse trigonometric functions

☛Definition
☛Range
☛domain
☛principal value branch
☛inverse trigonometric functions
☛properties of inverse trigonometric functions
☛Graphs of inverse trigonometric functions
☛Elementary properties of inverse trigonometric functions.

Inverse trigonometric functions Definition:

Inverse trigonometric functions also called as “anti-trigonometric” functions are nothing but the inverses of the basic trigonometric functions (i.e. sine, cosine, tangent, cotangent, secant, and cosecant) with suitable restricted domains.

Inverse trigonometric functions Notation

[latex display=”true”](i)sine inverse- \sin^{-1}x [/latex] [latex display=”true”](ii)cos inverse- \cos^{-1}x\cdot [/latex] [latex display=”true”](iii)tan inverse- \tan^{-1}x [/latex] [latex display=”true”](iv)cot inverse- cot^{-1}x[/latex] [latex display=”true”](v) secant inverse- Sec^{-1}x [/latex] [latex display=”true”](vi)cosecant inverse- \cos ec^{-1}x[/latex]

Range:

The range of the function is the set of outputs that can be generated by inputting the numbers from the domain into the function and perform the function operation. Range can also be the all real possible numbers(unlimited).

Function	Range
$f (x) = \sin^{- 1} (x)$	$[- \frac{π}{2}, \frac{π}{2}]$
$f (x) = \cos^{- 1} (x)$	$[0, π]$
$f (x) = \tan^{- 1} (x)$	$(- \frac{π}{2}, + \frac{π}{2})$
$f (x) = c o t^{- 1} (x)$	$(0, π)$
$f (x) = s e c^{- 1} (x)$	$[0, \frac{π}{2}) \cup [π, \frac{3 π}{2})$
$f (x) = \cos e c^{- 1} (x)$	$(- π, \frac{π}{2}] \cup (0, \frac{π}{2}]$

Domain:

The domain of a function is the set of possible values of the independent variable or variables of that function.

Function	Domain
$f (x) = \sin^{1} (x)$	$[- 1, 1]$
$f (x) = \cos^{- 1} (x)$	$[- 1, 1]$
$f (x) = \tan^{1} (x)$	$(- \infty, + \infty)$
$f (x) = c o t^{- 1} (x)$	$(- \infty, + \infty)$
$f (x) = s e c^{- 1} (x)$	$(- \infty, - 1] \cup [1, + \infty)$
$f (x) = \cos e c^{- 1} (x)$	$(- \infty, - 1] \cup [1, + \infty)$

Principal value branch:

Not even one inverse trigonometric function is one to one, Therefore, the ranges of the inverse functions are proper subsets of the domains of the original functions. when one function is desired to one value then the function may be restricted to its principal branch. With this restriction, the inverse of the sine function will evaluate to a single value, which is called the principal value.

The below-given table gives the principal value branches of the inverse trigonometric functions along with their domains and ranges.

Function	Domain	Range
$\sin^{- 1}$	$[- 1, 1]$	$[\frac{- π}{2}, \frac{π}{2}]$
$\cos^{- 1}$	$[- 1, 1]$	$[0, π]$
$\tan^{- 1}$	$R$	$(- \frac{π}{2}, \frac{π}{2})$
$c o t^{- 1}$	$R$	$(0, π)$
$S e c^{- 1}$	$R - (- 1, 1)$	$[0, π] - \{\frac{π}{2}\}$
$\cos e c^{- 1}$	$R - (- 1, 1)$	$[\frac{π}{2}, \frac{π}{2}] - \{0\}$

Inverse trigonometric functions:

sin to the power of negative 1 end exponent x

cos to the power of negative 1 end exponent x

Function	Domain	Range
$sin to the power of negative 1 end exponent x$	[−1,1]	[−π/2,π/2]
$cos to the power of negative 1 end exponent x$	[−1,1]	[0,π]
$tan to the power of negative 1 end exponent x$	(−∞,∞)	[−π/2,π/2]
$C o t to the power of negative 1 end exponent x$	(−∞,∞)	[0,π]
$S e c to the power of negative 1 end exponent x$	(−∞,−1]∪[1,∞)	[0,π/2)∪(π/2,π]
$cos e c to the power of negative 1 end exponent x$	(−∞,−1]∪[1,∞)	[−π/2,0)∪(0,π/2]

properties of inverse trigonometric functions:

1. Property number 1 [latex display=”true”](i). \sin^{-1}\frac1x=\cos ec^{-1}x,x\geq1\;or\;x\leq1[/latex] [latex display=”true”](ii). \cos^{-1}x\frac1x=sec^{-1}x,x\geq1\;or\;x\leq-1[/latex] [latex display=”true”](iii). \tan^{-1}x=\mathrm{co}t^{-1}x,x>0[/latex] 2. Property number 2 [latex display=”true”](i). Sin^{-1}\left(-x\right)=-\sin^{-1}x,\;x\in\lbrack-1,1\;\rbrack [/latex] [latex display=”true”](ii) \tan^{-1}\left(-x\right)=-\tan^{-1}x,\;x\in R[/latex] [latex display=”true”](iii) \cos ec^{-1}\left(-x\right)=-\cos ec^{-1}x,1x\vert\geq1[/latex] 3. Property number 3 [latex display=”true”](i) \cos^{-1}\left(-x\right)=\pi-\cos^{-1}x,\;x\in\lbrack-1,1\;\rbrack[/latex] [latex display=”true”](ii) Sec^{-1}\left(-x\right)=\pi-sec^{-1}x_,\;\left|x\right|\geq1[/latex] [latex display=”true”](iii) cot^{-1}\left(-x\right)=\pi-cot^{-1}x,\;x\in R [/latex] 4. Property number 4 [latex display=”true”](i) \sin^{-1}x+\cos^{-1}x=\frac\pi2,x\in\left[-1,1\right] [/latex] [latex display=”true”](ii) \tan^{-1}x+\cos^{-1}x=\frac\pi2,x\in R [/latex] [latex display=”true”](iii) \cos ec^{-1}x+sec^{-1}x=\frac\pi2,\;\left|x\geq1\right| [/latex] 5. Property number 5 [latex display=”true”](i) \tan^{-1}x+\tan^{-1}y=\tan^{-1}\frac{x+y}{1-xy},xy<1 [/latex] [latex display=”true”](ii) \tan^{-1}x-\tan^{-1}y=\tan^{-1}\frac{x-y}{1+xy},xy>-1 [/latex] [latex display=”true”](iii) \tan^{-1}x+\tan^{-1}y=\pi+\tan^{-1}\left(\frac{x+y}{1-xy}\right),\;xy>1;\;x,y>0 [/latex] 6. Property number 6 [latex display=”true”](i) 2\tan^{-1}x=\sin^{-1}\frac{2x}{1+x^2},\;\vert x\vert\leq1 [/latex] [latex display=”true”](ii) 2\tan^{-1}x=\cos^{-1}\frac{1-x^2}{1+x^2},x\geq0[/latex] [latex display=”true”](iii) 2\tan^{-1}x=\tan^{-1}\frac{2x}{1-x^2},-1<x<1[/latex]

Graphs of inverse trigonometric functions:

EXERCISE	SHORT ANSWERS	LONG ANSWERS	TOTAL SOLUTIONS
EXERCISE 2.1	14	0	14
EXERCISE 2.1	4	17	21