Trigonometric Functions Last Updated : 27 Feb, 2025 Comments Improve Suggest changes Like Article Like Report Trigonometric Functions, often simply called trig functions, are mathematical functions that relate the angles of a right triangle to the ratios of the lengths of its sides.Trigonometric functions are the basic functions used in trigonometry and they are used for solving various types of problems in physics, Astronomy, Probability, and other branches of science. There are six basic trigonometric functions used in Trigonometry which are:Sine Function (sin x)Cosine Function (cos x)Secant Function (tan x)Cosecant Function (cosec x)Tangent Function (tan x)Cotangent Function (cot x)Six Trigonometric FunctionsThe image added below shows a right-angle triangle PQR.Then the six basic trigonometric functions formulas for this right angle triangle are,FunctionSidesDescription Relationsin θPQ/PR Perpendicular/Hypotenusesin θ = 1/csc θcos θQR/PRBase/Hypotenusecos θ = 1/sec θtan θPQ/QRPerpendicular/Basetan θ = 1/cot θsec θPR/PQHypotenuse/Basesec θ = 1/cos θcosec θPR/QRHypotenuse/Perpendicularcosec θ = 1/sin θcot θQR/PQBase/Perpendicularcot θ = 1/tan θRead More: Trigonometric function RatiosValues of Trigonometric FunctionsThe value of trigonometric functions can easily be given using the trigonometry table. These values of the trigonometric functions are very useful in solving various trigonometric problems. The required trigonometry table is added below: The table added above shows all the values of the important angles from 0 to 180 degrees for all the trigonometric functions.Trigonometric Functions in Four(4) QuadrantsThe trigonometric functions are the periodic functions and their values repeat after a certain interval. Also, not all the trigonometric functions are positive in all the quadrants.An image explaining the same is added below:We divide the cartesian space into four quadrants namely, I, II, III, and IV quadrants, and the value of the trigonometric functions whether they are positive or negative in each quadrant is given as,I Quadrant: All PositiveII Quadrant: sin θ and cosec θ PositiveIII Quadrant: tan θ and cot θ PositiveIV Quadrant: cos θ and sec θ PositiveTrigonometric Functions GraphTrigonometric functions graphs plot the value of the trigonometric functions for different values of the angle(θ). For some the trigonometric functions are bounded as,Trigonometric functions sin θ and cos θ are bounded between - 1 and 1 and their graphs oscillate between -1 and 1 on the y-axis. Graph of the trigonometric function tan θ, and cot θ has a range from negative infinity to positive infinity.Graph of the trigonometric function sec θ, and cosec θ has a range from negative infinity to positive infinity excluding (-1, 1).Read More: Graph of Trigonometric FunctionsDomain and Range of Trigonometric FunctionsSuppose we have a trigonometric function f(x) = sin x, then the domain of the function f(x) is all the values of x that the function f(x) can take, and the domain is all possible outcomes of the f(x). The domain and range of all the six trigonometric functions are:Trigonometric FunctionDomainRangesin xR[-1, +1]cos xR[-1, +1]tan xR - (2n + 1)π/2Rcot xR - nπRsec xR - (2n + 1)π/2(-∞, -1] U [+1, +∞)cosec xR - nπ(-∞, -1] U [+1, +∞)Read in Detail- Domain and Range of Trigonometric Functions.Properties of Trigonometric FunctionsSome of the common properties of trigonometric functions are discussed below:Period refers to the length of one complete cycle of a trigonometric function, after which the function repeats.Sine (sin), Cosine (cos), Secant (sec), Cosecant (csc): Period = 2πTangent (tan), Cotangent (cot): Period = πSymmetry refers to the property that describes how the function behaves under reflection, translation, or rotation.Even Functions: f(−θ) = f(θ) (Cosine and Secant).Odd Functions: f(−θ) = −f(θ) (Sine, Tangent, Cosecant, Cotangent).Derivatives of Trigonometric FunctionsDifferentiation of trigonometric function can be easily found and the slope of that curve for that specific value of the trigonometric functions. The differentiation of all six trigonometric functions is added below:d/dx (sin x) = cos xd/dx (cos x) = -sin xd/dx (tan x) = sec2xd/dx (cot x) = -cosec2xd/dx (sec x) = sec x tan x d/dx (cosec x) = -cosec x cot xIntegration of Trigonometric FunctionsAs the integration of any curve gives the area under the curve, the integration of the trigonometric function also gives the area under the trigonometric function. The integration of various trigonometric functions is added below.∫ cos x dx = sin x + C∫ sin x dx = -cos x + C∫ tan x dx = log|sec x| + C∫ cot x dx = log|sin x| + C∫ sec x dx = log|sec x + tan x| + C∫ cosec x dx = log|cosec x - cot x| + CSome other important trigonometric integrals are:∫ sec2x dx = tan x + C∫ cosec2x dx = -cot x + C∫ sec x tan x dx = sec x + C∫ cosec x cot x dx = -cosec x + CRelated Reads:,Application of Trigonometry in Real LifeTrigonometric EquationsTrigonometric Symbols Comment More infoAdvertise with us Next Article Trigonometric Functions of Sum and Difference of Two Angles S salim25kkhan Follow Improve Article Tags : Mathematics School Learning Class 11 Trigonometry Trigonometry - MAQ Maths-Class-11 +2 More Similar Reads CBSE Class 11 Maths Notes CBSE Class 11 Maths Revision Notes have been designed in the most basic and detailed format possible, covering nearly all domains such as differential calculus, arithmetic, trigonometry, and coordinate geometry. We know how hard it gets when you shift to an altogether new grade where subjects are no 15+ min read Chapter 1: SetsRepresentation of a SetSets are defined as collections of well-defined data. In Math, a Set is a tool that helps to classify and collect data belonging to the same category. Even though the elements used in sets are all different from each other, they are all similar as they belong to one group. For instance, a set of dif 8 min read Types Of SetsIn mathematics, a set is defined as a well-defined collection of distinct elements that share a common property. These elements— like numbers, letters, or even other sets are listed in curly brackets "{ }" and represented by capital letters. For example, a set can include days of the week. The diffe 13 min read Universal SetsUniversal Set is a set that has all the elements associated with a given set, without any repetition. Suppose we have two sets P = {1, 3, 5} and Q = {2, 4, 6} then the universal set of P and Q is U = {1, 2, 3, 4, 5, 6}. We generally use U to denote universal sets. Universal Set is a type of set that 6 min read Venn DiagramVenn diagrams are visual tools used to show relationships between different sets. They use overlapping circles to represent how sets intersect, share elements, or stay separate. These diagrams help categorize items, making it easier to understand similarities and differences. In mathematics, Venn di 14 min read Operations on SetsSets are fundamental in mathematics and are collections of distinct objects, considered as a whole. In this article, we will explore the basic operations you can perform on sets, such as union, intersection, difference, and complement. These operations help us understand how sets interact with each 15+ min read Union of SetsUnion of two sets means finding a set containing all the values in both sets. It is denoted using the symbol '∪' and is read as the union. Example 1:If A = {1, 3. 5. 7} and B = {1, 2, 3} then A∪B is read as A union B and its value is,A∪B = {1, 2, 3, 5, 7}Example 2:If A = {1, 3. 5.7} and B = {2, 4} t 12 min read Chapter 2: Relations & FunctionsCartesian Product of SetsThe term 'product' mathematically refers to the result obtained when two or more values are multiplied together. For example, 45 is the product of 9 and 5.To understand the Cartesian product of sets, one must first be familiar with basic set operations such as union and intersection, which are appli 7 min read Relations and FunctionsIn mathematics, we often deal with sets of numbers or objects and the ways they are connected. Two important concepts that help us describe these connections are relations and functions.A relation is simply a connection between two sets of objects. Think of it as a rule that pairs elements from one 3 min read Domain and Range of RelationsThe domain is the set of all possible input values (the "x" values), and the range is the set of all possible output values (the "y" values) in a relation.For any two non-empty sets A and B, we define the relation R as the subset of the Cartesian product of A × B where each member of set A is relate 8 min read Piecewise FunctionPiecewise Function is a function that behaves differently for different types of input. As we know a function is a mathematical object which associates each input with exactly one output. For example: If a function takes on any input and gives the output as 3. It can be represented mathematically as 11 min read Range of a FunctionFunctions in math can be thought of as vending machines. Given the money in the form of input, they give some cans or cookies in return. Similarly, functions take some input numbers and give us some output. It can be said that, in real life, Everything can be formulated and solved with the help of f 7 min read Chapter 3: Trigonometric FunctionsMeasuring AnglesMeasurement of angles is very important in geometry for solving various geometrical problems and comparing angles in various geometrical figures. We use various geometric tools such as a protractor, and a compass for measuring angles. There are various types of angles measured by us and before learn 7 min read Trigonometric FunctionsTrigonometric Functions, often simply called trig functions, are mathematical functions that relate the angles of a right triangle to the ratios of the lengths of its sides.Trigonometric functions are the basic functions used in trigonometry and they are used for solving various types of problems in 6 min read Trigonometric Functions of Sum and Difference of Two AnglesTrigonometry 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 15+ min read Chapter 4: Principle of Mathematical InductionPrinciple of Mathematical InductionMathematical induction is a concept in mathematics that is used to prove various mathematical statements and theorems. The principle of mathematical induction is sometimes referred to as PMI. It is a technique that is used to prove the basic theorems in mathematics which involve the solution up to n 11 min read Chapter 5: Complex Numbers and Quadratic EquationsComplex NumbersComplex numbers are an essential concept in mathematics, extending the idea of numbers to include solutions for equations that don't have real solutions. Complex numbers have applications in many scientific research areas, signal processing, electromagnetism, fluid dynamics, quantum mechanics, and v 12 min read Algebra of Real FunctionsThe algebra of real functions refers to the study and application of algebraic operations on functions that map real numbers to real numbers. A function can be thought of as a rule or set of rules which map an input to an output knows as its image. It is represented as x ⇢ Function ⇢ y. A real funct 5 min read Algebraic Operations on Complex NumbersA complex number is a number that includes both a real and an imaginary part. It is written in the form:z = a + biWhere:a is the real part,b is the imaginary part,i is the imaginary unit, satisfying i2 = −1.Algebraic operations on complex numbers follow specific rules based on their real and imagina 7 min read Polar Representation of Complex NumbersComplex numbers, which take the form z = x + yi, can also be represented in a way that highlights their geometric properties. This alternative representation is known as the polar form. The polar representation of a complex number expresses it in terms of its magnitude (modulus) and direction (argum 9 min read Absolute Value of a Complex NumberThe absolute value (also called the modulus) of a complex number z = a + bi is its distance from the origin in the complex plane. The absolute value tells you how far a number is from zero, regardless of its direction (positive or negative).It is denoted as ∣z∣ and is given by the formula:|z| = \sqr 7 min read Conjugate of Complex NumbersIn the world of mathematics, complex numbers are one of the most important discoveries by mathematicians as they help us solve many real-life problems in various fields such as the study of electromagnetic waves, engineering, and physics.The Conjugate of a Complex Number is also a complex number obt 6 min read Imaginary NumbersImaginary numbers are numbers as the name suggests are the number that is not real numbers. All the numbers real and imaginary come under the categories of complex numbers. Imaginary numbers are very useful in solving quadratic equations and other equations whose solutions can not easily be found us 9 min read Chapter 6: Linear InequalitiesCompound InequalitiesCompound Inequalities are the combination of two or more inequalities. These inequalities are combined using two conditions that are AND, and OR. These conditions have specific meanings and they are solved differently. The inequities in compound inequalities are individually solved using normal rule 10 min read Algebraic Solutions of Linear Inequalities in One VariableA linear inequality is a mathematical expression involving an inequality symbol (<, >, ≤, or ≥) and a linear expression. Unlike linear equations, which give a specific solution, linear inequalities define a range of possible solutions.Example: 2x+3>5 In this case, the inequality indicates t 8 min read Graphical Solution of Linear Inequalities in Two VariablesWe know how to formulate equations of different degree, and it is used a lot in real life, but the question arises, is it always possible to convert a situation into an equation? Sometimes we get statements like, the number of Covid cases per day in Delhi has reached more than 10,000. This phrase “L 8 min read Solving Linear Inequalities Word ProblemsWe are well versed with equations in multiple variables. Linear Equations represent a point in a single dimension, a line in a two-dimensional, and a plane in a three-dimensional world. Solutions to linear inequalities represent a region of the Cartesian plane. It becomes essential for us to know ho 10 min read Chapter 7: Permutations and CombinationsFundamental Principle of CountingThe fundamental principle of counting is a basic concept used to determine the total number of possible outcomes in a situation where there are multiple independent events. It allows us to count a large number of possibilities without needing to list each one individually.For example, consider guess 11 min read PermutationIn Mathematics, Permutation is defined as a mathematical concept that determines the number of possible arrangements for a specific set of elements. therefore, it plays a big role in computer science, cryptography, and operations research. For example, take a set {1, 2, 3}:All Permutations taking al 15+ min read CombinationsCombination is a way of choosing items from a set, (unlike permutations) when the order of selection doesn't matter. In smaller cases, it's possible to count the number of combinations. Combination refers to the mixture of n things taken k at a time without repetition.Example: For set S = {a, b, c}, 8 min read Chapter 8: Binomial TheoremBinomial TheoremBinomial theorem is a fundamental principle in algebra that describes the algebraic expansion of powers of a binomial. According to this theorem, the expression (a + b)n where a and b are any numbers and n is a non-negative integer. It can be expanded into the sum of terms involving powers of a and 15+ min read Pascal's TrianglePascal's Triangle is a triangular arrangement of numbers where each number is the sum of the two numbers directly above it in the previous row. It is named after the French mathematician Blaise Pascal, although it was studied by mathematicians in various cultures long before him.Various patterns can 9 min read Chapter 9: Sequences and SeriesSequences and SeriesA sequence is an ordered list of numbers following a specific rule. Each number in a sequence is called a "term." The order in which terms are arranged is crucial, as each term has a specific position, often denoted as an​, where n indicates the position in the sequence.For example:2, 5, 8, 11, 14, 10 min read General and Middle Terms - Binomial Theorem - Class 11 MathsBinomial theorem or expansion describes the algebraic expansion of powers of a binomial. According to this theorem, it is possible to expand the polynomial "(a + b)n" into a sum involving terms of the form "axzyc", the exponents z and c are non-negative integers where z + c = n, and the coefficient 7 min read Arithmetic SeriesAn arithmetic series is the sum of the terms of an arithmetic sequence, where an arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. Or we can say that an arithmetic progression can be defined as a sequence of numbers in which for every pair of 5 min read Arithmetic SequenceAn arithmetic sequence or progression is defined as a sequence of numbers in which the difference between one term and the next term remains constant.For example: the given below sequence has a common difference of 1.1 2 3 4 5 . . . n ⇑ ⇑ ⇑ ⇑ ⇑ . . . 1st 2nd 3rd 4th 5th . . . nth TermsThe Arithmetic 8 min read Geometric Progression or GPGeometric Progression (GP) is a sequence of numbers where each term after the first is found by multiplying the previous term by a constant called the common ratio.For Example, the sequence given below forms a GP with a common ratio of 2 1 2 4 8 16 . . . n⇑ ⇑ ⇑ ⇑ ⇑ . . . 1st 2nd 3rd 4th 5th . . . nt 12 min read Geometric SeriesIn a Geometric Series, every next term is the multiplication of its Previous term by a certain constant, and depending upon the value of the constant, the Series may increase or decrease.Geometric Sequence is given as: a, ar, ar2, ar3, ar4,..... {Infinite Sequence}a, ar, ar2, ar3, ar4, ....... arn { 3 min read Arithmetic Progression and Geometric ProgressionArithmetic Progression and Geometric Progression: The word "sequence" in English means a collection of some numbers or objects in such a way that it has a first member, a second member, and so on. Sequences can be of anything, for example. - January, February, .... is the sequence of months in a yea 10 min read Special Series in Maths - Sequences and Series | Class 11 MathsSpecial Series: A series can be defined as the sum of all the numbers of the given sequence. The sequences are finite as well as infinite. In the same way, the series can also be finite or infinite. For example, consider a sequence as 1, 3, 5, 7, … Then the series of these terms will be 1 + 3 + 5 + 10 min read Like