Trigonometry is a branch of mathematics, which deals with the angles, lengths, and heights of triangles and their relationships. It had played an important role in calculating complex functions or large distances which were not possible to calculate without trigonometry. While solving problems with trigonometry, we came across many situations where we had to calculate the trigonometric solutions for the sum of angles or differences of angles.
Table of Contents
The following equalities in trigonometry will be used in the upcoming discussion to establish a relation between the sum and difference of angles
cos (-x) = cos x
sin (-x) = -sin x
We will now focus on the trigonometric functions which involve the sum and difference of two angles.
Trigonometric functions of sum and difference of angles
Consider the following figure:
A circle is drawn with center as origin and radius 1 unit. A point P1 is chosen at an angle of x units from x-axis. The co-ordinates are mentioned in the figure. Another point P2 is chosen, at an angle of y units from the line segment OP1. P3 is a point on the circle which is at an angle of y units from x-axis, measured clockwise.
Now, in the given figure, Δ OP1P3 is congruent to Δ OP2P4, by SAS congruency criteria.
Hence, P1P3 = P2P4 (CPCT)
⇒(P1P3 )2= (P2P4)2
Since we know the coordinates of all the four points, hence using distance formula, we can write:
[cos x – cos (-y)]2 + [sin x – sin (-y)]2 = [1- cos (x+y)]2 + sin2 (x+y)
On solving the above equation, we have the following identity:
cos(x + y) = cos x cos y – sin x sin y ……… (1)
Replacing y by -y in identity (1), we get,
cos(x – y) = cos x cos y + sin x sin y …….… (2)
Also,
cos (π/2 – x) = sin x ……………… (3)
That can be obtained by replacing x by π/2 and y by x in identity (2). Also,
sin (π/2 – x) = cos x………….…… (4)
As, sin (π/2 – x) = cos [π/2 – (π/2 – x)] (using identity 3). So,
sin (π/2 – x) = cos x
Now we have the idea about the expansion of sum and difference of angles of cos. Now let us try to use it for finding the values of sum and difference of angles of sin.
sin (x + y) can be written as cos [π/2 – (x + y)] which is equal to cos [(π/2 – x) – y]
Now, using identity (2) we can write,
cos [(π/2 – x) – y] = cos (π/2 – x) cos y + sin (π/2 – x) sin y
= sin x cos y + cos x sin y
Hence,
sin (x + y) = sin x cos y + cos x sin y …………………………. (5)
Replace y by –y in the above formula, we get
sin (x – y) = sin x cos y – cos x sin y .…………………………….. (6)
Now if we substitute suitable values in identities (1), (2), (5) and (6), we have the following:
cos (π/2 + x) = -sin x
sin (π/2 + x) = cos x
cos (π± x) = – cos x
sin (π – x) = sin x
sin (π + x) = – sin x
sin (2π – x) = -sin x
cos (2π – x) = cos x
After having a brief idea about the expansion of sum and difference of angles of sin and cos, the expansion for tan and cot is given by
tan (x + y) = (tan x + tan y)/ (1-tan x tan y)
tan (x – y) = (tan x – tan y)/ (1+tan x tan y)
Similarly;
cot (x + y) = (cot x cot y – 1)/(cot y + cot x)
cot (x – y) = (cot x cot y + 1)/(cot y – cot x)
What is the sum formula for sine?
The sum formula for sine is: sin(A+B)=sinAcosB+cosAsinBsin(A+B)=sinAcosB+cosAsinB
How can the difference formula for cosine be written?
The difference formula for the cosine is: cos(A–B)=cosAcosB+sinAsinBcos(A–B)=cosAcosB+sinAsinB
Why are these trigonometric identities important?
These identities simplify the computation of the trigonometric functions involving the multiple angles and are essential in solving various trigonometric equations.
How can the tangent of the sum of the two angles be expressed?
The tangent of the sum of the two angles can be expressed as:tan(A+B)=tanA+tanB1–tanAtanBtan(A+B)=1–tanAtanBtanA+tanB
Can these formulas be used for any angles A and B?
Yes, these formulas are valid for the any angles A and B.
Sum and Difference of Two Angles in Trigonometry
Trigonometric Identities for Sum and Difference of Angles
JEE Important Questions on Sum and Difference of Angles
Trigonometric Identities
Class 11 Math
JEE Study Material
Trigonometry Formulas
Important Questions
Trigonometric Equations
Trigonometry Notes
JEE Preparation
Trigonometry PDF
Math Formulas
Trigonometric Functions
Trigonometry Problems
JEE Main Trigonometry
Trigonometry Basics
Trigonometric Ratios
Trigonometry for Competitive Exams
Trigonometry Study Material
Trigonometry Practice Questions
Trigonometry Concepts
Trigonometry Cheat Sheet
Neeraj Anand, Param Anand
Er. Neeraj K.Anand is a freelance mentor and writer who specializes in Engineering & Science subjects. Neeraj Anand received a B.Tech degree in Electronics and Communication Engineering from N.I.T Warangal & M.Tech Post Graduation from IETE, New Delhi. He has over 30 years of teaching experience and serves as the Head of Department of ANAND CLASSES. He concentrated all his energy and experiences in academics and subsequently grew up as one of the best mentors in the country for students aspiring for success in competitive examinations.
In parallel, he started a Technical Publication "ANAND TECHNICAL PUBLISHERS" in 2002 and Educational Newspaper "NATIONAL EDUCATION NEWS" in 2014 at Jalandhar. Now he is a Director of leading publication "ANAND TECHNICAL PUBLISHERS", "ANAND CLASSES" and "NATIONAL EDUCATION NEWS".
He has published more than hundred books in the field of Physics, Mathematics, Computers and Information Technology. Besides this he has written many books to help students prepare for IIT-JEE and AIPMT entrance exams. He is an executive member of the IEEE (Institute of Electrical & Electronics Engineers. USA) and honorary member of many Indian scientific societies such as Institution of Electronics & Telecommunication Engineers, Aeronautical Society of India, Bioinformatics Institute of India, Institution of Engineers. He has got award from American Biographical Institute Board of International Research in the year 2005.
CBSE Class 11 Maths Syllabus for 2023-24 with Marking Scheme
CBSE syllabus for class 11 Maths is divided into 5 units. The table below shows the units, number of periods and marks allocated for maths subject. The maths theory paper is of 80 marks and the internal assessment is of 20 marks.
No.
Units
Marks
I.
Sets and Functions
23
II.
Algebra
25
III.
Coordinate Geometry
12
IV.
Calculus
08
V.
Statistics and Probability
12
Total Theory
80
Internal Assessment
20
Grand Total
100
2025-26 CBSE Class 11 Maths Syllabus
Below you will find the CBSE Class Maths Syllabus for students.
Unit-I: Sets and Functions
1. Sets
Sets and their representations, empty sets, finite and infinite sets, equal sets, subsets, and subsets of a set of real numbers, especially intervals (with notations), universal set, Venn diagrams, union and intersection of sets, difference of sets, complement of a set and properties of complement.
2. Relations & Functions
Ordered pairs, Cartesian product of sets, number of elements in the Cartesian product of two finite sets, Cartesian product of the set of reals with itself (upto R x R x R), definition of relation, pictorial diagrams, domain, co-domain and range of a relation. Function as a special type of relation. Pictorial representation of a function, domain, co-domain and range of a function. Real valued functions, domain and range of these functions, constant, identity, polynomial, rational, modulus, signum, exponential, logarithmic and greatest integer functions, with their graphs. Sum, difference, product and quotients of functions.
3. Trigonometric Functions
Positive and negative angles, measuring angles in radians and in degrees and conversion from one measure to another, definition of trigonometric functions with the help of unit circle, truth of the identity, signs of trigonometric functions, domain and range of trigonometric functions and their graphs, expressing sin (x±y) and cos (x±y) in terms of sinx, siny, cosx & cosy and their simple applications.
Unit-II: Algebra
1. Complex Numbers and Quadratic Equations
Need for complex numbers, especially√−1, to be motivated by the inability to solve some of the quadratic equations. Algebraic properties of complex numbers, Argand plane.
2. Linear Inequalities
Linear inequalities, algebraic solutions of linear inequalities in one variable and their representation on the number line.
3. Permutations and Combinations
The fundamental principle of counting. Factorial n. (n!) Permutations and combinations, derivation of Formulae for nPr and nCr and their connections, simple applications.
4. Binomial Theorem
Historical perspective, statement and proof of the binomial theorem for positive integral indices, Pascal’s triangle, simple applications.
5. Sequence and Series
Sequence and series, arithmetic progression (A. P.), arithmetic mean (A.M.), geometric progression (G.P.), general term of a G.P., sum of n terms of a G.P., infinite G.P. and its sum, geometric mean (G.M.), relation between A.M. and G.M.
Unit-III: Coordinate Geometry
1. Straight Lines
Brief recall of two-dimensional geometry from earlier classes. Slope of a line and angle between two lines. Various forms of equations of a line: parallel to axis, point-slope form, slope-intercept form, two-point form, intercept form and normal form. General equation of a line. Distance of a point from a line.
2. Conic Sections
Sections of a cone: circles, ellipse, parabola, hyperbola, a point, a straight line and a pair of intersecting lines as a degenerated case of a conic section. Standard equations and simple properties of parabola, ellipse and hyperbola. Standard equation of a circle.
3. Introduction to Three-Dimensional Geometry
Coordinate axes and coordinate planes in three dimensions. Coordinates of a point. Distance between two points.
Unit-IV: Calculus
1. Limits and Derivatives
Derivative introduced as rate of change both as that of distance function and geometrically, intuitive idea of limit, limits of polynomials and rational functions trigonometric, exponential and logarithmic functions, definition of derivative relate it to the slope of the tangent of the curve, derivative of sum, difference, product and quotient of functions. Derivatives of polynomial and trigonometric functions.
Unit-V: Statistics and Probability
1. Statistics
Measures of Dispersion: Range, mean deviation, variance and standard deviation of ungrouped/grouped data.
2. Probability
Events; occurrence of events, ‘not’, ‘and’ and ‘or’ events, exhaustive events, mutually exclusive events, Axiomatic (set theoretic) probability, connections with other theories of earlier classes. Probability of an event, probability of ‘not’, ‘and’ and ‘or’ events.
Students can also get the syllabus of all the subjects by visiting CBSE Class 11 Syllabus page. Learn Maths & Science in an interactive & fun-loving way with Anand Classes App/Tablet.
Frequently Asked Questions on CBSE Class 11 Maths Syllabus 2025-26
Q1
What is the marks distribution for internals and theory exams according to the CBSE Maths Syllabus for Class 11?
The marks distribution for internals is 20 marks and the theory exam is 80 marks based on the CBSE Class 11 Maths Syllabus.
Q2
Which is the most important chapter in the CBSE Class 11 Maths Syllabus?
The important chapter in the CBSE Class 11 Maths Syllabus is Algebra which is for 25 marks in the overall weightage.
Q3
What are the chapters covered in Unit III of the CBSE Class 11 Maths Syllabus?
The chapters covered in Unit III of the CBSE Class 11 Maths Syllabus are straight lines, conic sections and an introduction to three-dimensional geometry.
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