# 3.15: Inverse Trig Derivatives Exercises - Mathematics

### Exercise:

For the following exercises, use the graph of (y=f(x)) to

a. sketch the graph of (y=f^{−1}(x)), and

b. use part a. to estimate ((f^{−1})′(1)).

261)

a.

b. ((f^{−1})′(1)~2)

262)

263)

a.

b. ((f^{−1})′(1)~−1/sqrt{3})

For the following exercises, use the functions (y=f(x)) to find

a. (frac{df}{dx}) at (x=a) and

b. (x=f^{−1}(y).)

c. Then use part b. to find (frac{df^{−1}}{dy}) at (y=f(a).)

264) (f(x)=6x−1,x=−2)

265) (f(x)=2x^3−3,x=1)

(a. 6
b. x=f^{−1}(y)=(frac{y+3}{2})^{1/3}
c. frac{1}{6})

266) (f(x)=9−x^2,0≤x≤3,x=2)

267) (f(x)=sin x,x=0)

(a. 1, b. x=f^{−1}(y)=sin^{−1}y, c. 1)

For each of the following functions, find ((f^{−1})′(a)).

268) (f(x)=x^2+3x+2,x≥−1,a=2)

269) (f(x)=x^3+2x+3,a=0)

(frac{1}{5})

270) (f(x)=x+sqrt{x},a=2)

271) (f(x)=x−frac{2}{x},x<0,a=1)

frac{1}{3})

272) (f(x)=x+sin x,a=0)

273) (f(x)= an x+3x^2,a=0)

(1)

For each of the given functions (y=f(x),)

a. find the slope of the tangent line to its inverse function (f^{−1}) at the indicated point (P), and

b. find the equation of the tangent line to the graph of (f^{−1}) at the indicated point.

274) (f(x)=frac{4}{1+x^2},P(2,1))

275) (f(x)=sqrt{x−4},P(2,8))

(a. 4, b. y=4x)

276) (f(x)=(x^3+1)^4,P(16,1))

277) (f(x)=−x^3−x+2,P(−8,2))

(a. −frac{1}{96}, b. y=−frac{1}{13}x+frac{18}{13})

278) (f(x)=x^5+3x^3−4x−8,P(−8,1))

For the following exercises, find (frac{dy}{dx}) for the given function.

279) (y=sin^{−1}(x^2))

(frac{2x}{sqrt{1−x^4}})

280) (y=cos^{−1}(sqrt{x}))

281) (y=sec^{−1}(frac{1}{x}))

(frac{−1}{sqrt{1−x^2}})

282) (y=sqrt{csc^{−1}x})

283) (y=(1+ an^{−1}x)^3)

(frac{3(1+ an^{−1}x)^2}{1+x^2})

284) (y=cos^{−1}(2x)⋅sin^{−1}(2x))

285) (y=frac{1}{ an^{−1}(x)})

(frac{−1}{(1+x^2)( an^{−1}x)^2})

286) (y=sec^{−1}(−x))

287) (y=cot^{−1}sqrt{4−x^2})

(frac{x}{(5−x^2)sqrt{4−x^2}})

288) (y=x⋅csc^{−1}x)

For the following exercises, use the given values to find ((f^{−1})′(a)).

289) (f(π)=0,f'(π)=−1,a=0)

(−1)

290) (f(6)=2,f′(6)=frac{1}{3},a=2)

291) (f(frac{1}{3})=−8,f'(frac{1}{3})=2,a=−8)

(frac{1}{2})

292) (f(sqrt{3})=frac{1}{2},f'(sqrt{3})=frac{2}{3},a=frac{1}{2})

293) (f(1)=−3,f'(1)=10,a=−3)

(frac{1}{10})

294) (f(1)=0,f'(1)=−2,a=0)

295) [T] The position of a moving hockey puck after (t) seconds is (s(t)= an^{−1}t) where (s) is in meters.

a. Find the velocity of the hockey puck at any time (t).

b. Find the acceleration of the puck at any time (t).

c. Evaluate a. and b. for (t=2,4),and (6) seconds.

d. What conclusion can be drawn from the results in c.?

a. (v(t)=frac{1}{1+t^2})
b. (a(t)=frac{−2t}{(1+t^2)^2})
c. ((a) v(2)=0.2, v(4)=frac{1}{17},v(6)=frac{1}{37};(b)a(2)=−0.16,a(4)=−frac{8}{289},a(6)=−frac{12}{1369})
d. The hockey puck is decelerating/slowing down at 2, 4, and 6 seconds.

296) [T] A building that is 225 feet tall casts a shadow of various lengths (x) as the day goes by. An angle of elevation (θ) is formed by lines from the top and bottom of the building to the tip of the shadow, as seen in the following figure. Find the rate of change of the angle of elevation (frac{dθ}{dx}) when (x=272) feet.

297) [T] A pole stands 75 feet tall. An angle (θ) is formed when wires of various lengths of (x) feet are attached from the ground to the top of the pole, as shown in the following figure. Find the rate of change of the angle (frac{dθ}{dx}) when a wire of length 90 feet is attached.

298) [T] A television camera at ground level is 2000 feet away from the launching pad of a space rocket that is set to take off vertically, as seen in the following figure. The angle of elevation of the camera can be found by (θ= an^{−1}(frac{x}{2000})), where x is the height of the rocket. Find the rate of change of the angle of elevation after launch when the camera and the rocket are 5000 feet apart.

299) [T] A local movie theater with a 30-foot-high screen that is 10 feet above a person’s eye level when seated has a viewing angle (θ) (in radians) given by (θ=cot^{−1}frac{x}{40}−cot^{−1}frac{x}{10}),

where (x) is the distance in feet away from the movie screen that the person is sitting, as shown in the following figure.

a. Find (frac{dθ}{dx}).

b. Evaluate (frac{dθ}{dx}) for (x=5,10,15,) and 20.

c. Interpret the results in b..

d. Evaluate (frac{dθ}{dx}) for (x=25,30,35), and 40

e. Interpret the results in d. At what distance (x) should the person stand to maximize his or her viewing angle?

a. (frac{dθ}{dx}=frac{10}{100+x^2}−frac{40}{1600+x^2} b. frac{18}{325},frac{9}{340},frac{42}{4745},0) c. As a person moves farther away from the screen, the viewing angle is increasing, which implies that as he or she moves farther away, his or her screen vision is widening. d. (−frac{54}{12905},−frac{3}{500},−frac{198}{29945},−frac{9}{1360} e. As the person moves beyond 20 feet from the screen, the viewing angle is decreasing. The optimal distance the person should stand for maximizing the viewing angle is 20 feet.

## Inverse Trigonometric Derivatives

In this lesson, we will look at how to find the derivatives of inverse trigonometric functions.

### Table Of Derivatives Of Inverse Trigonometric Functions

The following table gives the formula for the derivatives of the inverse trigonometric functions. Scroll down the page for more examples and solutions on how to use the formulas.

Example:
Differentiate

Solution:
We can use the above formula and the chain rule.

Example:
Differentiate

Solution:
We use the product rule and chain rule.

Inverse Trigonometric Functions - Derivatives

Formulas for the derivatives of the six inverse trig functions and derivative examples.

Examples:
Find the derivatives of the following functions

Inverse Trigonometric Functions - Derivatives - Harder Example

Example:
Find the derivatives of
y = sec -1 √(1 + x 2 )

Inverse Trigonometric Functions - Derivatives - Harder Example

Example:
Find the derivatives of
y = sin -1 (cos x/(1+sinx))

Derivatives of Inverse Trig Functions

One example does not require the chain rule and one example requires the chain rule.

Examples:
Find the derivatives of each given function.

Derivatives of Inverse Trig Functions

Examples:
Find the derivatives of each given function.

Try the free Mathway calculator and problem solver below to practice various math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.

## 3. Geometry & Trigonometry

The Geometry & Trigonometry topic has twice as much content at the HL level (51 suggested teaching hrs) than the SL level (25 suggested teaching hrs) because there is no vectors content at the SL level.

exercise sets, quizzes & tests for teaching units in Geometry & Trig

10 Feb 2021: New trigonometry unit test that can be used with SL and/or HL students and full worked solutions has been posted: AA_SLHL_Test1_trig

Changes compared to previous Maths HL-SL syllabusses :
As mentioned above, the most significant change is that vectors is only at HL level. Thus, at SL level it is expected that when a student is required to determine the measure of an angle it wil always involve the use of right triangle trigonometry. The only other change is the addition of the following content in syllabus item SL 3.1*: volume and surface area of 3D solids including right pyramid, right cone, sphere, hemisphere and combinations of these solids.

Geometry & Trigonometry - syllabus overview

Syllabus item numbers are in brackets.

### SL - Geometry & Trig

SL core (AA & AI)

triangle trigonometry & applications (3.2, 3.3)

angles, circles, arcs & sectors (3.4)

trig functions, equations & identities (3.5 - 3.8)

sine rule &ndash ambiguous case (3.5)

### HL - Geometry & Trig

further trig functions, identities & properties (3.9 - 3.11)

scalar product & applications (3.13)

vectors &ndash equations of lines (3.14, 3.15)

vector product & applications (3.16)

lines and planes in space (3.17, 3.18)

### Exercise Sets, Quizzes & Tests

AA_SL_3.2(8)_trig1_v1
Set of six questions covering trigonometry content from syllabus item 3.2 to 3.8. No GDC on Qs 1-4 GDC allowed on Qs 5 & 6. The syllabus content is appropriate for both SL and HL students. Worked solutions are included on the 2nd page.

AA_SL_Test1_trig_v1
Trigonometry test covering the following syllabus items: using trigonometric ratios to find sides and angles of right-angled triangles area of triangle area of sector angles of elevation & depression sine rule (including ambiguous case) cosine rule use of bearings angle between a line and a plane. Solution key (worked solutions) available below.

AA_SL_Test1_trig_v1_SOL_KEY
Solution key (worked solutions) for SL trigonometry test above.

AA_SLHL_Test1_trig_v1
Trigonometry unit test that can be used with SL and/or HL students. Questions 1-7 for SL students (40 marks), and Qs 1-9 for HL students (50 marks). Questions 8 & 9 are good 'discriminating' questions for HL students that is, students for which HL is too challenging will usually not do well on Qs 8 & 9. Solution key (worked solutions) available below.

AA_SLHL_Test1_trig_v1_SOL_KEY
Solution key (worked solutions) for SL-HL trigonometry unit test above.

## Implicit Differentiation

which are the top and bottom halves of a cricle respectively, to define the functional relation completely.

 And xy = sin( y )+ x 2 y 2 (the magenta curves in the figure at the left) cannot be solved for either y as an explicit function of x or x as an explicit function of y . This implicit function is considered in Example 2.

Perhaps surprisingly, we can take the derivative of implicit functions just as we take the derivative of explicit functions. We simply take the derivative of each side of the equation, remembering to treat the dependent variable as a function of the independent variable, apply the rules of differentiation, and solve for the derivative. Returning to our original example:

This is of course what we get from differentiating the explicit form, y = 2 x -3, with respect to x .

This simple example may not be very enlightening. Consider the second example, the equation that describes a circle of radius 3, centered at the origin. Taking the derivative of both sides with respect to x , using the power rule for the derivative of y ,

It can be seen from the figures that for either part of the circle, the slope of the tangent line has the opposite sign of the ratio x / y , and that the magnitude of the slope becomes larger as the tangent point nears the x -axis.

(For this case, finding dy/dx as an explicit function of x requires use of the power rule for fractional powers, usually considered later. This example may be thought of as a taste of things to come.)

### Some examples:

Note that this expression can be solved to give x as an explicit function of y by solving a cubic equation, and finding y as an explicit function of x would involve soving a quartic equation, neither of which is in our plan.

Using the chain rule and treating y as an implicit function of x ,

In this case, the chain rule and product rule are both used to advantage:

The use of inverse trigonometric functions allow this to be solve for y as an explicit function of x and graphed, as shown. However, this function serves as a good example of implicit differentiation:

## Integration Worksheets

Integration worksheets include basic integration of simple functions, integration using power rule, substitution method, definite integrals and more.

Good practice sheets for calculus beginners. Learn the rule of integrating functions and apply it here.

Integrate Using Power Rule

If dy/dx = x n , then after integration y = x n+1 / n+1 + C, where C is integral constant.

Substitution Method of Integration

Set the numerator or denominator as different variable (depends on compatibility), differentiate, substitute in appropriate place, rewrite, and then integrate.

Definite Integral Worksheets

Definite integral is a basic tool in application of integration. Finding the value of the function between the x values graphically represents the area of the function under the curve within the x limits.

## NCERT Solutions Class 12 Maths Chapter 2 Inverse Trigonometric Function

### Maths Part I

1. Relations and Functions
2. Inverse Trigonometric Functions
3. Matrices
4. Determinants
5. Continuity and Differentiability
6. Application of Derivatives

### Maths Part II

1. Integrals
2. Application of Integrals
3. Differential Equations
4. Vector Algebra
5. Three Dimensional Geometry
6. Linear Programming
7. Probability

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## 3.15: Inverse Trig Derivatives Exercises - Mathematics

NCERT Solutions for Class 12 Maths

Solutions for all the questions from Maths NCERT, Class 12th

DIFFERENT TYPES OF MAPPINGS (FUNCTIONS)

DIFFERENT TYPES OF MAPPINGS (FUNCTIONS)

DIFFERENT TYPES OF MAPPINGS (FUNCTIONS)

Doubtnut is India’s leading e-learning app for students of class 12th. Get here the solution of every question mentioned in your NCERT Maths book from chapter 1 – Relation Functions to Chapter 12 Linear programming. These self-explanatory video tutorials are prepared by the experts to offer homework help, and learning assistance to study smart. We provide free and fully-solved Class 12 Maths NCERT Solutions for you learn the Maths concept effectively. NCERT solutions for class 12 Maths is based on the latest NCERT syllabus and following the CCE guidelines by CBSE.

Below are the chapter-wise 12 links given for your NCERT textbook. Select the chapter you wish to study and follow the expert guidance through the video tutorial. All the chapters are sub-divided into individual exercise. The solutions are as per the latest CCE marking scheme. These 5-10 minutes of pre-recorded NCERT tutorials are meant for your thorough understanding of each & every NCERT topic. Open any 12th class NCERT Maths Solutions video and you will find an expert guiding your way. With the convenience of learning in HD prime quality. Simply take notes from these tutorials religiously to score best grades in your CBSE board Maths examination. Also, you can anytime rewind or fast-forward the videos as per your comfort and pace of learning.

Cengage Chapterwise Maths Solutions

INVERSE TRIGONOMETRIC FUNCTIONS

CONTINUITY AND DIFFERENTIABILITY

APPLICATION OF DERIVATIVES

THREE DIMENSIONAL GEOMETRY

Key Highlights of NCERT Solutions for Class 12 Maths:

1. All Maths video solutions in Hinglish (Hindi-English).

2. Step by step solutions for a better understanding of concepts.

3. Sequential exercise-wise solutions of all the questions of NCERT Maths textbook.

4. Maths tips and tricks to provide an in-depth understanding of concepts.

5. PDF solutions for Class 12 Maths where all the solutions are arranged chapter-wise.

6. With these video tutorials, you can study anytime-anywhere

Short and Fun Videos Lessons for Class 12 Maths NCERT Solutions

Mathematics is not only about learning. It comes through the regular practice of all the questions given in your NCERT Maths textbooks. These NCERT video lessons are prepared by the expert Mathematicians to provide with better learning assistance towards achieving higher grades. NCERT Maths Solutions for Class 12 covers all the exercises given in Chapters: Relations and Functions, Inverse Trigonometric Functions, Matrices, Determinants, Continuity and Differentiability, Application of Derivatives, Integrals, Application of Integrals, Differential Equations, Vector Algebra, Three Dimensional Geometry, Linear Programming, Probability.

Chapter 1: Relations and Functions

Provided here are detailed step-by-step NCERT Solutions for Chapter 1 – Relations and Functions. Here you’ll find the basics of relations and functions - definitions, types and more! It is an important chapter in class 12 for CBSE students. Class 12th Maths Chapter 1 Relations and Functions NCERT Solution is given below. Exercise 1.1 Introductions Exercise 1.2 Types of Relations Exercise 1.3 Types of Functions Exercise 1.4 Compositions of Functions and Invertible Function Exercise 1.5 Binary Operations

Chapter 2: Inverse Trigonometric Functions

In this lesson, you’ll be learning about the inverse trigonometric functions, also known as the arcus functions or anti trigonometric functions. Precisely, they are the inverse functions of other trigonometric functions such as the sine, cosine, tangent, and are used to obtain an angle from any of the angle’s trigonometric ratios. This chapter has an important application in engineering, physics, and geometry. The exercise-wise solutions we’ve covered include: Exercise 2.1 Introduction Exercise 2.2 Basic Concepts Exercise 2.3 Properties of Inverse Trigonometric Functions

In chapter 3 of class 12 maths, you'll be learning what matrices are and they're all sort of uses including solving systems of equations, transforming shapes and vectors, and representing real-world situations, etc. You’ll also be learning about addition, subtraction and multiplication of the matrices, and how to find the inverses of matrices. The exercise wise NCERT solutions we’ve covered are listed below: Exercise 3.1 Introduction Exercise 3.2 Matrix Exercise 3.3 Types of Matrices Exercise 3.4 Operations on Matrices Exercise 3.5 Transpose of a Matrix Exercise 3.6 Symmetric and Skew-Symmetric Matrices Exercise 3.7 Elementary Operation (Transformation) of a Matrix Exercise 3.8 Invertible Matrices

A determinant is an extremely important quantity that we calculate from a matrix. In chapter 4, you’ll be learning how to compute determinants, its properties, and verifications. We’ve covered all the exercise of this chapter and also provided the NCERT exemplar solutions as well. The exercises included in the Determinants chapter are the following: Exercise 4.1 Introduction Exercise 4.2 Determinant Exercise 4.3 Properties of Determinants Exercise 4.4 Area of a Triangle Exercise 4.5 Adjoint and Inverse of a Matrix Exercise 4.6 Applications of Determinants and Matrices Exercise 4.7 Summary

Chapter 5: Continuity and Differentiability

NCERT Solutions class 12 Maths Chapter 5 Continuity and Differentiability are available in video tutorial format for free. These NCERT exercise wise solutions are very helpful for students to do their homework on time and prepare for CBSE board exam. Listed below are the exercise wise solutions: Exercise 5.1 Introduction Exercise 5.2 Algebra of continuous functions Exercise 5.3 Differentiability Exercise 5.4 Derivatives of composite functions Exercise 5.5 Derivatives of implicit functions Exercise 5.6 Derivatives of inverse trigonometric functions Exercise 5.7 Exponential and Logarithmic Functions Exercise 5.8 Logarithmic Differentiation Exercise 5.9 Derivatives of Functions in Parametric Forms Exercise 5.10 Second Order Derivative Exercise 5.11 Mean Value Theorem Exercise 5.12 Summary

Chapter 6: Application of Derivatives

The derivative is an expression that gives the rate of change of a function w.r.t. a self-determining variable. This chapter has important applications in Mathematics as well as other domains of science and engineering. You will be learning about applications of derivatives in the exercises mentioned below. Exercise 6.1 Introduction Exercise 6.2 Rate of Change of Quantities Exercise 6.3 Increasing and Decreasing Functions Exercise 6.4 Tangents and Normals Exercise 6.5 Approximations Exercise 6.6 Maxima and Minima Exercise 6.7 Maximum and Minimum Values of a Function in a Closed Interval Exercise 6.8 Summary

In this chapter, you’ll be learning about the Integrals in-depth. Starting from the exercise 7.1 introduction to integrals, you’ll be learning how to do integration as an inverse process of differentiation. then you’ll learn about the method of integration, followed by the integrals of some particular functions, integration by partial fractions, integration by parts and the definite integral. in exercise 7.8 you'll be introduced to the fundamental theorem of calculus and finally in exercise 7.9 will be about the evaluation of definite integrals by substitution Exercise-wise solutions we’ve covered include: Exercise 7.1 Introduction Exercise 7.2 Integration as an Inverse Process of Differentiation Exercise 7.3 Methods of Integration Exercise 7.4 Integrals of some Particular Functions Exercise 7.5 Integration by Partial Fractions Exercise 7.6 Integration by Parts Exercise 7.7 Definite Integral Exercise 7.8 Fundamental Theorem of Calculus Exercise 7.9 Evaluation of Definite Integrals by Substitution Exercise 7.10 Some Properties of Definite Integrals

Chapter 8: Application of Integrals

In chapter 8 you will learn about the applications of the integrals. This section covers areas under simple curves and area between the two curves. You will also be learning how to find the area and volume of the curved figures. The exercise wise solutions we’ve covered are: Exercise 8.1 Introduction Exercise 8.2 Area under Simple Curves Exercise 8.3 Area between Two Curves

Chapter 9 : Differential Equations

In chapter 9, you will be learning some basic concepts related to the differential equation. This will introduce to general and particular solutions of a differential equation, formation of differential equations, first-degree differential equation and some applications of differential equations. We’ve covered all 5 important exercises in this chapter. With given NCERT solutions student can study the Differential equations chapter in a detailed manner. Exercise 9.1 Introduction Exercise 9.2 Basic Concepts Exercise 9.3 General and Particular Solutions of a Differential Equation Exercise 9.4 Formation of a Differential Equation whose General Solution is given Exercise 9.5 Methods of Solving First order, First Degree Differential Equations

Chapter 10: Vector Algebra

In Chapter 10 Vector Algebra, you will be learning about some basic concepts of vectors. We’ve provided the complete video tutorial on each and every concept related to this chapter including the various operations on vectors, and their algebraic as well as geometric properties. The exercise wise solutions are listed below: Exercise 10.1 Introduction Exercise 10.2 Some Basic Concepts Exercise 10.3 Types of Vectors Exercise 10.4 Addition of Vectors Exercise 10.5 Multiplication of a Vector by a Scalar Exercise 10.6 Product of Two Vectors

Chapter 11: Three Dimensional Geometry

In chapter 11 three dimensional geometry you will learn about the study of the direction cosines and direction ratios of a line, equations of a line in space, the angle between the two lines, the shortest distance between the two lines and plans and planes, etc. The video tutorials provide step-by-step solutions related to this chapter. The exercise wise NCERT solutions that you’ll find here are listed below: Exercise 11.1 Introduction Exercise 11.2 Direction Cosines and Direction Ratios of a Line Exercise 11.3 Equation of a Line in Space Exercise 11.4 Angle between Two Lines Exercise 11.5 Shortest Distance between Two Lines Exercise 11.6 Plane Exercise 11.7 Coplanarity of Two Lines Exercise 11.8 Angle between Two Lines Exercise 11.9 Distance of a Point from a P Exercise 11.10 Angle between a Line and a Plane

Chapter 12: Linear Programming

NCERT Solutions for Linear Programming is prepared by the best mathematician and 12th board examination experts. The video tutorial includes all the important topics from very exercise. Each answer comes with an in-depth explanation to help students this chapter better. The NCERT solutions are provided in a step by step manner for your better understanding. The exercises included in the Linear Programming chapter are the following: 12.1 Introduction 12.2 Linear Programming Problem and its Mathematical Formulation 12.3 Different Types of Linear Programming Problems

Probability is the last chapter of NCERT class 12th maths books. In this chapter, you’ll be introduced to conditional probability, multiplication theorem on probability, independent events, Bayes’ theorem, random variables, and its probability distributions and Bernoulli trials and the binomial distribution. The video tutorials cover every exercise in this chapter in the most comprehensive manner. Listed below are the exercise wise solutions that we’ve covered: Exercise 13.1 Introduction Exercise 13.2 Conditional Probability Exercise 13.3 Multiplication Theorem on Probability Exercise 13.4 Independent Events Exercise 13.5 Bayes’ Theorem Exercise 13.6 Random Variables and its Probability Distributions Exercise 13.7 Bernoulli Trials and Binomial Distribution

Doubtnut also facilitates downloadable chapter-wise PDF solutions for class 12 Maths. Our aim is to help the student achieve better grades in board examination and score highest possible rank in the competitive exams including IIT-JEE, BITSAT, VITEEE, SRMJEEE, and others… Also, check out our 4.4/5 rated education App ‘Doubtnut’ on the Google Play Store. The best App in the play store for Mathematics practice and learning

## NCERT Class 12 Mathematics Chapter-wise Solutions

### Part I

• Chapter 1: Relations and Functions
• Chapter 2: Inverse Trigonometric Functions
• Chapter 3: Matrices
• Chapter 4: Determinants
• Chapter 5: Continuity and Differentiability

### Part II

• Chapter 6: Application of Derivatives
• Chapter 7: Integrals
• Chapter 8: Application of Integrals
• Chapter 9: Differential Equations
• Chapter 10: Vector Algebra
• Chapter 11: Three-Dimensional Geometry
• Chapter 12: Linear Programming
• Chapter 13: Probability

CBSE class 12th Mathematics have two books. Each book has chapters and topics.

1. NCERT Mathematics Book Class 12 Part-1
2. NCERT Mathematics Book Class 12 Part-2

Here is the list of topics covered under each chapter of class 12 Mathematics NCERT text book.

### 1. Relations and Functions

• 1.1 Introduction
• 1.2 Types of Relations
• 1.3 Types of Functions
• 1.4 Composition of Functions and Invertible Function
• 1.5 Binary Operations

### 2. Inverse Trigonometric Functions

• 2.1 Introduction
• 2.2 Basic Concepts
• 2.3 Properties of Inverse Trigonometric Functions

### 3. Matrices

• 3.1 Introduction
• 3.2 Matrix
• 3.3 Types of Matrices
• 3.4 Operations on Matrices
• 3.5 Transpose of a Matrix
• 3.6 Symmetric and Skew Symmetric Matrices
• 3.7 Elementary Operation (Transformation) of a Matrix
• 3.8 Invertible Matrices

### 4. Determinants

• 4.1 Introduction
• 4.2 Determinant
• 4.3 Properties of Determinants
• 4.4 Area of a Triangle
• 4.5 Minors and Cofactors
• 4.6 Adjoint and Inverse of a Matrix
• 4.7 Applications of Determinants and Matrices

### 5. Continuity and Differentiability

• 5.1 Introduction
• 5.2 Continuity
• 5.3 Differentiability
• 5.4 Exponential and Logarithmic Functions
• 5.5 Logarithmic Differentiation
• 5.6 Derivatives of Functions in Parametric Forms
• 5.7 Second Order Derivative
• 5.8 Mean Value Theorem

### 6. Application of Derivatives

• 6.1 Introduction
• 6.2 Rate of Change of Quantities
• 6.3 Increasing and Decreasing Functions
• 6.4 Tangents and Normal
• 6.5 Approximations
• 6.6 Maxima and Minima

### 7. Integrals

• 7.1 Introduction
• 7.2 Integration as an Inverse Process of Differentiation
• 7.3 Methods of Integration
• 7.4 Integrals of some Particular Functions
• 7.5 Integration by Partial Fractions
• 7.6 Integration by Parts
• 7.7 Definite Integral
• 7.8 Fundamental Theorem of Calculus
• 7.9 Evaluation of Definite Integrals by Substitution
• 7.10 Some Properties of Definite Integrals

### 9. Differential Equations

• 9.1 Introduction
• 9.2 Basic Concepts
• 9.3 General and Particular Solutions of a Differential Equation
• 9.4 Formation of a Differential Equation whose General Solution is given
• 9.5 Methods of Solving First order, First Degree Differential Equations

### 10. Vector Algebra

• 10.1 Introduction
• 10.2 Some Basic Concepts
• 10.3 Types of Vectors
• 10.5 Multiplication of a Vector by a Scalar
• 10.6 Product of Two Vectors

### 11. Three Dimensional Geometry

• 11.1 Introduction
• 11.2 Direction Cosines and Direction Ratios of a Line
• 11.3 Equation of a Line in Space
• 11.4 Angle between Two Lines
• 11.5 Shortest Distance between Two Lines
• 11.6 Plane
• 11.7 Coplanarity of Two Lines
• 11.8 Angle between Two Planes
• 11.9 Distance of a Point from a Plane
• 11.10 Angle between a Line and a Plane

12. Linear Programming

• 12.1 Introduction
• 12.2 Linear Programming Problem and its Mathematical Formulation
• 12.3 Different Types of Linear Programming Problems

13. Probability

• 13.1 Introduction
• 13.2 Conditional Probability
• 13.3 Multiplication Theorem on Probability
• 13.4 Independent Events
• 13.5 Bayes' Theorem
• 13.6 Random Variables and its Probability Distributions
• 13.7 Bernoulli Trials and Binomial Distribution

## 6.3 Inverse Trigonometric Functions

For any right triangle , given one other angle and the length of one side, we can figure out what the other angles and sides are. But what if we are given only two sides of a right triangle? We need a procedure that leads us from a ratio of sides to an angle. This is where the notion of an inverse to a trigonometric function comes into play. In this section, we will explore the inverse trigonometric functions .

### Understanding and Using the Inverse Sine, Cosine, and Tangent Functions

In order to use inverse trigonometric functions, we need to understand that an inverse trigonometric function “undoes” what the original trigonometric function “does,” as is the case with any other function and its inverse. In other words, the domain of the inverse function is the range of the original function, and vice versa, as summarized in Figure 1.

In previous sections, we evaluated the trigonometric functions at various angles, but at times we need to know what angle would yield a specific sine, cosine, or tangent value. For this, we need inverse functions. Recall that, for a one-to-one function , if f ( a ) = b , f ( a ) = b , then an inverse function would satisfy f − 1 ( b ) = a . f − 1 ( b ) = a .

Bear in mind that the sine, cosine, and tangent functions are not one-to-one functions. The graph of each function would fail the horizontal line test. In fact, no periodic function can be one-to-one because each output in its range corresponds to at least one input in every period, and there are an infinite number of periods. As with other functions that are not one-to-one, we will need to restrict the domain of each function to yield a new function that is one-to-one. We choose a domain for each function that includes the number 0. Figure 2 shows the graph of the sine function limited to [ − π 2 , π 2 ] [ − π 2 , π 2 ] and the graph of the cosine function limited to [ 0 , π ] . [ 0 , π ] .

Figure 3 shows the graph of the tangent function limited to ( − π 2 , π 2 ) . ( − π 2 , π 2 ) .

These conventional choices for the restricted domain are somewhat arbitrary, but they have important, helpful characteristics. Each domain includes the origin and some positive values, and most importantly, each results in a one-to-one function that is invertible. The conventional choice for the restricted domain of the tangent function also has the useful property that it extends from one vertical asymptote to the next instead of being divided into two parts by an asymptote.

On these restricted domains, we can define the inverse trigonometric functions .