NCERT Solutions for Class 11 Maths Chapter 3 Trigonometric Functions (Ex 3.4)

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Mathematics is a vitally important system of knowledge in human life and society. It has a distinctly scientific methodology, which implies the necessity of calculations and numerical verifications in order to prove postulates and reach logical conclusions. Mathematics has a long history which can be traced back to early human civilisations like the Babylonian and Indus Valley civilisations. This underlines the fact that Mathematics is a very important applied science and, by extension, a vital academic discipline. For the students of the CBSE Board who are pursuing their senior secondary education in the PCM stream, the NCERT textbook of Mathematics is a mandatory resource for learning.

For the optimum use of the potential of this fundamental text, the NCERT Solutions prepared by the Extramarks platform for online learning could prove to be of great help. The Class 11 Maths NCERT Solutions Chapter 3 Exercise 3.4 have been prepared through a careful and thorough analysis of the prescribed syllabus for Mathematics by the NCERT. The Class 11 Maths NCERT Solutions Chapter 3 Exercise 3.4 are modelled on the basis of the structure of contents published in the NCERT textbook of Mathematics for Class 11.

NCERT Solutions for Class 11 Maths Chapter 3 Trigonometric Functions (Ex 3.4) Exercise 3.4

The Class 11 Maths NCERT Solutions Chapter 3 Exercise 3.4 have been prepared after extensive and earnestly committed research of the past years’ question papers and sample question papers.This was done to ensure that students could effectively use the Class 11 Maths NCERT Solutions Chapter 3 Exercise 3.4 to familiarise themselves with the pattern of mark distribution in board examinations.The Class 11 Maths NCERT Solutions Chapter 3 Exercise 3.4 have been fashioned in a framework that is easy to understand. Access NCERT Solutions for Class 11 Maths Chapter 3 – Trigonometric Functions

The Extramarks website has gone to extra lengths to craft the Class 11 Maths NCERT Solutions Chapter 3 Exercise 3.4 in a format which is very compatible with internet usage. The Class 11 Maths NCERT Solutions Chapter 3 Exercise 3.4 have been compiled by Extramarks with attentive consideration of the requirements and needs of students. Teachers of mathematics could use the Class 11 Maths NCERT Solutions Chapter 3 Exercise 3.4 to encourage students to self-study.Introduction to Trigonometry

Trigonometry refers to that branch of the academic discipline of Mathematics that deals with particular functions of angles and their applications to calculations. In layman’s language, it is that stream of Mathematics that is concerned with the relationship between the sides and angles of triangles.

The chronology of Trigonometry as a mathematical sub-discipline could be tracedback to the third century BC when astronomical studies were being conducted through the application of Geometry. The knowledge covered within the sub-discipline of Trigonometry could be practically applied in many ventures in professional and academic spheres such as architecture, surveying, astronomy, physics, engineering, and even crime scene investigations.Morris Kline explained in his renowned work, Mathematical Thought From Ancient to Modern Times, that while trigonometry first appeared in connection with astronomy, it was also used in navigation and the creation of calendars.The foundations of Trigonometry lie in Geometry.The Class 11 Maths NCERT Solutions, Chapter 3, Exercise 3.4, are online learning resources that are highly relevant and can address a variety of common student concerns.Students can use the Class 11 Maths NCERT Solutions Chapter 3 Exercise 3.4 to take personal notes, which can greatly aid in the retention of important information such as formulas, theorems, definitions, and so on.

Terms Used For Trigonometric Functions:

According to common usage, there are six functions of an angle which are utilised in Trigonometry. Their names and abbreviated forms are as follows: sine (sin), cosine (cos), tangent (tan), cotangent (cot), secant (sec) and cosecant (csc). Trigonometric functions are indispensable for discovering unknown angles and distances from known or measured angles in geometrical illustrations.

The Class 11 Maths NCERT Solutions Chapter 3 Exercise 3.4 contain all of the solutions to the exercises presented in the third chapter of the prescribed textbook for Mathematics for students in Class 11.Extramarks has been particularly attentive to providing step-by-step calculations as a part of the Class 11 Maths NCERT Solutions Chapter 3 Exercise 3.4. This would make it easier and more feasible for students to follow a logical sequence of problem solving and verifying obtained conclusions.

Trigonometric Equations

Those equations which include the application and usage of trigonometric functions like sin, cos, tan, cot, etc., are defined as trigonometric equations.

The Class 11 Maths NCERT Solutions Chapter 3 Exercise 3.4 could prove to be quite helpful to students in laying a reliable foundation with regard to the theme of Trigonometry which is a crucial field of study in many arenas of practical life. As a result, Extramarks’ Class 11 Maths NCERT Solutions Chapter 3 Exercise 3.4 could be used as high-quality reference materials by students preparing to compete in various competitive examinations.The NCERT Solutions For Class 11 Maths Chapter 3 Exercise 3.4 are available on the Extramarks educational website in a PDF version, which could be downloaded for future reference.

Theorems Discussed For Trigonometric Functions Exercise 3.4:

The Class 11 Maths NCERT Solutions Chapter 3 Exercise 3.4 have been solved in a detailed and comprehensive way. The formulae applied for solving the exercises provided as part of the Class 11 Maths NCERT Solutions Chapter 3 Exercise 3.4 have been highlighted separately. This has been done so that students would be able to follow the appropriate procedure necessary for reaching an adequate solution through a logically processed series of calculations.

Trigonometric Functions Exercise 3.4:

The Class 11 Maths NCERT Solutions, Chapter 3, Exercise 3.4, are versatile online learning resources with a wide range of significanceFor instance, the Class 11 Maths NCERT Solutions Chapter 3 Exercise 3.4 authored by Extramarks, have been equipped with adequate illustrations and diagrams wherever necessary. This has been done in order to ensure the availability of highly explained and descriptive solutions to the exercises which have been prescribed as a part of the NCERT textbook of Mathematics for Class 11.

Tips to Solve NCERT Solutions of Exercise 3.4 Class 11 Maths Chapter 3

Teachers may find it hard to address the doubts, concerns, and queries of students within the limited schedule of classroom teaching. Students’ questions and doubts may go unanswered which may have a detrimental effect on their conceptual clarity and comprehension abilities and consequently affect their performance in board examinations.

The Class 11 Maths NCERT Solutions Chapter 3 Exercise 3.4 includes solutions which have been cautiously curated and modelled by Extramarks in collaboration with experienced mentors and instructors of Mathematics with outstanding expertise in the discipline. As a result, teachers could use the Class 11 Maths NCERT Solutions Chapter 3 Exercise 3.4 as highly recommended online teaching tools for a versatile method of teaching.NCERT Solutions for Class 11 Maths Chapters [Include Chapter wise Pages]The Class 11 Maths NCERT Solutions Chapter 3 Exercise 3.4 are based on a simplified rendering of the academic curriculum encapsulated within the third chapter of the prescribed Mathematics textbook for Class 11 students.

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NCERT Solutions Class 11Maths of Chapter 3 All Exercises

The Class 11 Maths NCERT Solutions Chapter 3 Exercise 3.4 could be referenced by teachers. In the contemporary era, the pedagogies of classroom teaching are transforming with the aid of technology and innovation. Online teaching and smart classrooms have become the normsince the advent of technology. In such a rapidly changing context, it is crucial for teachers to be up-to-date with novel teaching and learning tools and techniques. The Class 11 Maths NCERT Solutions Chapter 3 Exercise 3.4 is one such modern teaching innovation which could also be of great advantage to students. Students would benefit greatly from the Class 11 Maths Chapter 3 Exercise 3.4 in their self-learning.About Extramarks:

With the availability of the Class 11 Maths NCERT Solutions Chapter 3 Exercise 3.4 on the Extramarks online portal for learning, students could receive expert guidance anytime. These solutions have been designed for all academic disciplines and for students and learners of all age groups. Extramarks also provides reference materials like solutions, past years’ papers, sample question papers, revision notes, chapter notes, and solutions for the course material of various educational boards across the country. The entirety of the scholarly curriculum being made available by the Extramarks educational website, like the Class 11 Maths NCERT Solutions Chapter 3 Exercise 3.4 is available in PDF format. This implies that content like the Class 11 Maths NCERT Solutions Chapter 3 Exercise 3.4 could be conveniently downloaded for future reference. The Extramarks website strongly reiterates its claim that with the aid of these solutions available for Class 11 Maths Chapter 3 Exercise 3.4, students would pass through board examinations with flying colours.

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The Extramarks website for online learning and education is a holistic destination for attaining accessibility to excellent academic services, which are the outcome of sincere efforts and painstaking research. To help students maximise their potential, the Extramarks online education platform offers online tutoring as well as reference materials for various types of academic content.The Extramarks learning portal also supplies reference materials such as NCERT Solutions for all prescribed course books for students of all age groups and educational boards which are a part of the educational setup under the purview of the National Council for Research and Training, or NCERT. Extramarks is a portal sincerely dedicated to the rendering of excellent education services within the country.

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Q.1

$\begin{array}{l} Find the principal and general solutions of the following \\ equations : \tan x = \sqrt{3} \end{array}$

Ans.

$\begin{array}{l} \begin{array}{l} Since, tan \frac{π}{3} = \sqrt{3} and tan \frac{4 π}{3} = tan (π + \frac{π}{3}) = \tan \frac{π}{3} = \sqrt{3} \\ Therefore, principal solutions are x = \frac{π}{3} and \frac{4 π}{3} . \\ General solution of tanx = \sqrt{3} \\ tanx = \sqrt{3} \\ = \tan \frac{π}{3} \end{array} \\ \begin{array}{l} tanx = \tan (nπ + \frac{π}{3}) \\ \Rightarrow x = nπ + \frac{π}{3}, where n \in Z . \end{array} \end{array}$

Q.2

$\begin{array}{l} Find the principal and general solutions of the following \\ equation : \sec x = 2 \end{array}$

Ans.

$\begin{array}{l} Since, secx = 2 \Rightarrow \sec \frac{π}{3} = 2 and sec \frac{5 π}{3} = \sec (2 π - \frac{π}{3}) = \sec \frac{π}{3} = 2 \\ Therefore, principal solutions are x = \frac{π}{3} and \frac{5 π}{3} . \\ G eneral solution: \\ secx = 2 \\ = \sec \frac{π}{3} \\ = \sec (2 nπ \pm \frac{π}{3}) \\ ∴ x = 2 nπ \pm \frac{π}{3}, where n \in Z . \end{array}$

Q.3

$\begin{array}{l} Find the principal and general solutions of the following \\ equation : \cot x = - \sqrt{3} \end{array}$

Ans.

$\begin{array}{l} Since, cot x = - \sqrt{3} \\ \Rightarrow cot x = - \cot \frac{π}{6} \\ = \cot (π - \frac{π}{6}) \\ = \cot (\frac{5 π}{6}) \\ and \\ cot x = \cot (2 π - \frac{π}{6}) \\ = \cot (\frac{11 π}{6}) \\ Therefore, the principal solution are x = \frac{5 π}{6} and \frac{11 π}{6} . \\ General solution: \\ cotx = - \sqrt{3} \\ = \cot (\frac{5 π}{6}) \\ = \cot (nπ + \frac{5 π}{6}) \\ \Rightarrow x = nπ + \frac{5 π}{6}, where n \in Z . \end{array}$

Q.4

$\begin{array}{l} Find the principal and general solutions of the following \\ equation : cosec x = - 2 \end{array}$

Ans.

$Since, cosec x = cosec (\frac{π}{6})$

$\begin{array}{l} ∴ cosec x = - 2 \\ = - cosec (\frac{π}{6}) \\ = cosec (π + \frac{π}{6}) \\ = cosec (\frac{7 π}{6}) \\ and \\ cosec x = cosec (2 π - \frac{π}{6}) \\ = cosec \frac{11 π}{6} \\ Therefore, the principal solutions are \frac{7 π}{6}, \frac{11 π}{6} . \\ General solution: \\ cosec x = cosec (\frac{7 π}{6}) \\ = cosec (nπ + {(- 1)}^{n} \frac{7 π}{6}) \\ \Rightarrow x = nπ + {(- 1)}^{n} \frac{7 π}{6}, n \in Z . \end{array}$

Q.5

$\begin{array}{l} Find the general solution for each of the following \\ equation : \cos 4 x = \cos 2 x \end{array}$

Ans.

$\begin{array}{l} We have, \\ cos 4x = cos 2x \end{array}$

$\begin{array}{l} = \cos (2 nπ \pm 2 x) [\begin{array}{l} The general solution of cos θ = \cos α \\ \Rightarrow θ = 2 nπ \pm α \end{array}] \\ \Rightarrow 4 x = 2 nπ \pm 2 x \\ For (+) ve sign: \\ 4 x = 2 nπ + 2 x \\ 2 x = 2 nπ \Rightarrow x = nπ \\ For (-) ve sign: \\ 4 x = 2 nπ - 2 x \\ 6 x = 2 nπ \Rightarrow x = \frac{1}{3} nπ \\ The general solutions are x = \frac{1}{3} nπ and nπ, where n \in Z . \end{array}$

Q.6

$\begin{array}{l} Find the general solution for each of the following \\ equation : \cos 3 x + \cos x - \cos 2 x = 0 \end{array}$

Ans.

$\begin{array}{l} We have, \cos 3 x + \cos x - \cos 2 x = 0 \\ 2 \cos (\frac{3 x + x}{2}) \cos (\frac{3 x - x}{2}) - \cos 2 x = 0 \\ 2 \cos 2 x \cos x - \cos 2 x = 0 \\ \cos 2 x (2 \cos x - 1) = 0 \\ \Rightarrow \cos 2 x = 0 \Rightarrow 2 x = (2 n + 1) \frac{π}{2} \\ \Rightarrow x = (2 n + 1) \frac{π}{4}, n \in Z \end{array}$

$\begin{array}{l} or 2 cos x - 1 = 0 \Rightarrow \cos x = \frac{1}{2} = \cos \frac{π}{3} \\ \Rightarrow x = 2 nπ \pm \frac{π}{3}, n \in Z . \end{array}$

Q.7

$\begin{array}{l} Find the general solution for each of the following \\ equation : \sin 2 x + \cos x = 0 \end{array}$

Ans.

$\begin{array}{l} We have, \sin 2 x + \cos x = 0 \\ \Rightarrow 2 \sin x \cos x + \cos x = 0 \\ \Rightarrow \cos x (2 \sin x + 1) = 0 \\ Either \cos x = 0 \Rightarrow x = (2 n + 1) \frac{π}{2}, n \in Z . \\ or (2 \sin x + 1) = 0 \Rightarrow \sin x = - \frac{1}{2} \\ \Rightarrow \sin x = \sin \frac{7 π}{2} \\ \Rightarrow x = nπ + {(- 1)}^{n} \frac{7 π}{2}, n \in Z . \\ Thus, the general solution of given equation is: \\ nπ + {(- 1)}^{n} \frac{7 π}{2} or (2 n + 1) \frac{π}{2}, n \in Z . \end{array}$

Q.8

$\begin{array}{l} Find the general solution for each of the following \\ equation : \sec^{2} 2 x = 1 - \tan 2 x \end{array}$

Ans.

$\begin{array}{l} We have, \sec^{2} 2 x = 1 - \tan 2 x \\ \Rightarrow 1 + \tan^{2} 2 x = 1 - \tan 2 x \end{array}$

$\begin{array}{l} \Rightarrow \tan^{2} 2 x + \tan 2 x = 0 \\ \Rightarrow \tan 2 x (\tan 2 x + 1) = 0 \\ Either tan 2x = 0 \Rightarrow 2 x = nπ \\ \Rightarrow x = \frac{nπ}{2}, where n \in Z . \\ or \tan 2 x + 1 = 0 \Rightarrow \tan 2 x = - 1 \\ \Rightarrow \tan 2 x = - \tan \frac{π}{4} \\ = \tan (π - \frac{π}{4}) \\ \Rightarrow \tan 2 x = \tan \frac{3 π}{4} \\ \Rightarrow 2 x = nπ + \frac{3 π}{4} \\ \Rightarrow x = \frac{nπ}{2} + \frac{3 π}{8}, where n \in Z . \\ The general solution of the given equation is: \\ x = \frac{nπ}{2}, \frac{nπ}{2} + \frac{3 π}{8}, where n \in Z . \end{array}$

Q.9

$\begin{array}{l} Find the general solution for each of the following \\ equation : \sin x + \sin 3 x + \sin 5 x = 0 \end{array}$

Ans.

$\begin{array}{l} We have, \sin x + \sin 3 x + \sin 5 x = 0 \\ \Rightarrow \sin 5 x + \sin x + \sin 3 x = 0 \\ \Rightarrow 2 \sin (\frac{5 x + x}{2}) \cos (\frac{5 x - x}{2}) + \sin 3 x = 0 \end{array}$

$\begin{array}{l} \Rightarrow 2 \sin 3 x \cos 2 x + \sin 3 x = 0 \\ \Rightarrow \sin 3 x (2 \cos 2 x + 1) = 0 \\ Either sin3x = 0 \Rightarrow 3 x = nπ \\ \Rightarrow x = \frac{nπ}{3}, n \in Z . \\ or 2 cos 2x + 1 = 0 \Rightarrow \cos 2 x = - \frac{1}{2} \\ \Rightarrow 2 x = 2 nπ \pm \frac{2 π}{3} \\ \Rightarrow x = nπ \pm \frac{π}{3}, n \in Z . \\ Therefore, the general solution of the given equations is: \\ x = \frac{nπ}{3} or nπ \pm \frac{π}{3}, n \in Z . \end{array}$

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FAQs (Frequently Asked Questions)

1. Could students find Class 11 Maths NCERT Solutions Chapter 3 Exercise 3.4 conveniently online?

The Extramarks educational platform is a single trustworthy and reliable destination for effortlessly and immediately procuring access to the Class 11 Maths NCERT Solutions Chapter 3 Exercise 3.4. It has designed the Class 11 Maths NCERT Solutions Chapter 3 Exercise 3.4 with attentive and thorough research of the prescribed textbooks for Mathematics by the NCERT for the students of Class 11. These solutions have been compiled while keeping the nuances of the updated NCERT syllabus in mind and are available in a downloadable format for convenient learning.

2. What other kinds of NCERT Solutions are available on the Extramarks educational website?

In addition to the NCERT Solutions for the Class 11 Maths NCERT Solutions Chapter 3 Exercise 3.4, the Extramarks educational website for online learning also consists of adept and high-quality NCERT Solutions for all academic disciplines prescribed for students of all age groups. As a matter of fact, every one of these solutions is available in the form of PDFs and can be downloaded whenever needed. This is greatly suggested that the availability of the NCERT Solutions Class 12 and NCERT Solutions Class 11 strongly solidifies the commitment of Extramarks that students would be impeccably prepared with regard to the entire academic curriculum prescribed for these vital classes. Extramarks strongly reiterates that with the aid of these resources, students would perform exceedingly well in the board examinations of Class 12. With regard to similarly widespread concern, the availability of NCERT Solutions Class 10 and NCERT Solutions Class 9 is objected towards preparing students for the first board examinations of their school curriculum. Extramarks is also providing accessibility to NCERT Solutions Class 7 and NCERT Solutions Class 8 to make it more feasible for the students to deal with the versatility and vastness of the prescribed academic syllabus for these classes. The availability of NCERT Solutions Class 6 would also be of great help to students of this class within the CBSE board.

NCERT Solutions for Class 11 Maths Chapter 3 Trigonometric Functions (Ex 3.4)

NCERT Solutions for Class 11 Maths Chapter 3 Trigonometric Functions (Ex 3.4) Exercise 3.4

Terms Used For Trigonometric Functions:

Trigonometric Equations

Theorems Discussed For Trigonometric Functions Exercise 3.4:

Trigonometric Functions Exercise 3.4:

Tips to Solve NCERT Solutions of Exercise 3.4 Class 11 Maths Chapter 3

NCERT Solutions Class 11Maths of Chapter 3 All Exercises

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NCERT Solutions for Class 11 Maths Chapter 3 Trigonometric Functions (Ex 3.4)

NCERT Solutions for Class 11 Maths Chapter 3 Trigonometric Functions (Ex 3.4) Exercise 3.4

Terms Used For Trigonometric Functions:

Trigonometric Equations

Theorems Discussed For Trigonometric Functions Exercise 3.4:

Trigonometric Functions Exercise 3.4:

Tips to Solve NCERT Solutions of Exercise 3.4 Class 11 Maths Chapter 3

NCERT Solutions Class 11Maths of Chapter 3 All Exercises

Why Choose Us?

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FAQs (Frequently Asked Questions)

1. Could students find Class 11 Maths NCERT Solutions Chapter 3 Exercise 3.4 conveniently online?

2. What other kinds of NCERT Solutions are available on the Extramarks educational website?

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NCERT Solutions

Solved Board Papers

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