# Trigonometric Function Formulas

## Trigonometric Function Formulas

A trigonometric function’s sign is determined by the signs of the coordinates of the points on the angle’s terminal side. The signs of all the Trigonometric Function Formulas can be found by determining which quadrant an angle’s terminal side belongs in. An angle’s terminal side can be found in any of the four quadrants or along the axes in either the positive or negative direction. These are the eight possible locations (the quadrantal angles). The signs of the Trigonometric Function Formulas have different meanings depending on the circumstance. The x and y coordinates’ signs can be either positive or negative, but the distance from a point to the origin is always positive. Thus, each of the six Trigonometric Function Formulas has a positive value in the first quadrant, where the x and y coordinates are all positive. Only sine and cosecant, which is sine’s reciprocal, are positive in the second quadrant. Only the tangent and cotangent are positive in the third quadrant. Finally, only cosine and secant are positive in the fourth quadrant. The values of the Trigonometric Function Formulas are either 0, 1, -1, or undefined when an angle is perpendicular to an axis. A trigonometric function’s undefined value indicates that division by zero was required to calculate the ratio for that particular function.

## What are Trigonometric Functions?

The branch of Mathematics known as trigonometry is responsible for identifying and analysing the relationships between triangle sides and the angles they subtend. Right-angled triangles primarily require the use of trigonometry. The connections between a triangle’s three sides and its three angles are described by Trigonometric Function Formulas. Generally, there are six Trigonometric Function Formulas. It is necessary to learn about the three sides of a right-angled triangle before moving on to the study of Trigonometric Function Formulas. Only acute angles are defined in the earliest definitions of trigonometric functions as they relate to right-angle triangles. Geometrical definitions using the standard unit circle, or a circle with a radius of one unit, are frequently used to extend the sine and cosine functions to functions whose domain is the entire real line; the real line’s isolated points are then removed from the domain of the other functions. Trigonometric Function Formulas are now expressed in modern definitions as infinite series or as the answers to differential equations. With some isolated points removed, this enables the extension of the domain of the sine and cosine functions to the entire complex plane as well as the domain of the other Trigonometric Function Formulas to the complex plane.

There are two equivalent ways to define Trigonometric Function Formulas in calculus: either by using differential equations or power series. These definitions are equivalent because it is simple to obtain the other as a property starting from either one of them. However, the definition using differential equations seems more logical because, for instance, the choice of the power series coefficients may seem to be made quite arbitrarily, and because it is much simpler to derive the Pythagorean identity from the differential equations.

### Trigonometric Functions Formulas

These series’ radii of convergence are infinite. Since entire functions (also known as “sine” and “cosine”) are complex-valued functions that are defined and holomorphic on the entire complex plane, the sine and the cosine can be extended to entire functions.

The other Trigonometric Function Formulas can be extended to meromorphic functions, which are holomorphic throughout the entire complex plane with the exception of a few isolated points known as poles, since they are defined as fractions of whole functions.

A right-angled triangle has the following three sides:

Base -The base is the side on which the angle ϴ is located.

Perpendicular – The side that is being considered is the one across from the angle ϴ.

Hypotenuse – In a right-angled triangle, it is the longest side and sits on the other side of the 90° angle.

Triangle-Based Functions

The six fundamental Trigonometric Function Formulas in trigonometry are sine, cosine, tangent, cosecant, and cotangent.

### Trigonometric Functions Values

Sine – The ratio of the perpendicular to the hypotenuse is known as sin, and it is represented by the symbol sin.

Cosine – The ratio of the base and hypotenuse is denoted by the symbol cos.

Tangent- The ratio of an angle’s sine and cosine is known as tan and is denoted by the symbol. As a result, the perpendicular to base ratio is the definition of a tangent.

Cosecant – It is referred to as cosec and is the opposite of sin.

Secant – The symbol for it, sec, is the reciprocal of cos.

Cotangent – It is denoted by the symbol cot and is the reciprocal of the number tan.

### Trig Functions in Four Quadrants

The plane is divided into four quadrants by the coordinate axes, which are First, Second, Third, and Fourth. For instance, angles in the third quadrant range from 180° to 270°. The sign of each trigonometric ratio in a particular quadrant can be determined by looking at the x and y coordinates of the point P as it lies in each of the four quadrants.

### Trigonometric Functions Graph

The domain value of ϴ is represented on the horizontal x-axis of Trigonometric Function Formulas graphs, and the range value is represented along the vertical y-axis. While the graphs of other Trigonometric Function Formulas do not pass through the origin, the graphs of Sin ϴ  and Tan ϴ do. Sin ϴ and Cos ϴ have a finite range of [-1, 1]. In quadrant 1 all the Trigonometric Function Formulas are positive. Only sine and cosecant are positive in quadrant 2. Tangent and cotangent have positive values in quadrant 3. In quadrant 4, cosine and secant are positive.

If an equation involving trigonometric ratios of an angle is true for all values of the angle, it is referred to as having a trigonometric identity. These come in handy whenever an expression or equation contains a Trigonometric Function Formulas. The sine, cosine, tangent, cosecant, secant, and cotangent are the six fundamental trigonometric ratios. The sides of the right triangle, such as the adjacent side, opposite side, and hypotenuse side, are used to define each of these trigonometric ratios.

### Domain and Range of Trigonometric Functions

The angle ϴ and the resultant value, respectively, define the domain and range of Trigonometric Function Formulas. Angles in degrees or radians are the domain of Trigonometric Function Formulas, and a real number is the range. Depending on the area where the trigonometric function is not defined, some values are excluded from the domain and range of Trigonometric Function Formulas.

### Trigonometric Functions Identities

Trigonometric identities are equality statements involving Trigonometric Function Formulas that are true for all possible values of the variables, defining both sides of the equality. Sin, cos, and tan are the three fundamental trigonometric ratios. The reciprocals of sin, cos, and tan are represented by the three additional trigonometric ratios sec, cosec, and cot in trigonometry, respectively.

### Reciprocal Identities

The reciprocals of the six primary Trigonometric Function Formulas—sine, cosine, tangent, cotangent, secant, and cosecant—are known as reciprocal identities. Reciprocal identities are not the same as inverse Trigonometric Function Formulas and this is an important distinction to make. The reciprocal of one trigonometric function is a property of every fundamental trigonometric function. For instance, the sine function’s reciprocal identity is cosecant.

### Pythagorean Identities

As the name implies, Pythagorean identities are derived from the Pythagoras theorem. This theorem states that the square of the hypotenuse, or longest side, in any right-angled triangle, equals the sum of the squares of the other two sides (legs). Trigonometric ratios (as they are defined for a right-angled triangle) can be applied using this theorem to produce Pythagorean identities. In Physics, Trigonometric Function Formulas are also crucial. The movement of a mass attached to a spring and small angles are examples of simple harmonic motion that mimics many natural phenomena.The pendular motion of a mass hanging by a string, is described by the sine and cosine functions, respectively. Uniform circular motion is projected in one dimension by the sine and cosine functions.

Additionally useful in the study of general periodic functions are Trigonometric Function Formulas. Modeling recurring phenomena like sound or light waves is made possible by the distinctive wave patterns of periodic functions.

### Sum and Difference Identities

Trigonometric functions are calculated using the sum and difference formulas at particular angles where it is simpler to express the angle as the sum or difference of distinct angles (0°, 30°, 45°, 60°, 90°, and 180°). Trigonometric function values at 0°, 30°, 45°, 60°, 90°, and 180° are memorized. So, in order to simplify the problem and determine the value of a trigonometric function at 105°, we can write it as 105° = 45° + 60°.

### Half-Angle Identities

The half-angle formula in trigonometry is used to calculate the precise trigonometric ratios of angles like 15° (half of the standard angle 30°), 22.5° (half of the standard angle 45°), and so forth. If is an angle, then /2 represents the half-angle. The Trigonometric Function Formulas sine, cosine, tangent, cosecant, secant, and cotangent are all well known. We will learn the values for these functions’ standard angles, such as 0°, 30°, 45°, 60°, and 90°, with the aid of the trigonometry table. In those Trigonometric Function Formulas that contain half angles, trigonometric half angle identities or functions are actually present. The presence of an angle in a quadrant completely determines whether the square root of the first two functions, sine and cosine, is positive or negative.

### Double Angle Identities

Trigonometric ratios of double angles (2ϴ) are expressed in terms of trigonometric ratios of single angles (ϴ) using double angle formulas. The Pythagorean identities are used to derive some alternative formulas, and the double angle formulas are special cases of (and are thus derived from) the sum formulas of trigonometry.

### Triple Angle Identities

It is important to learn all the triple-angle identities. Learning them will help students get a deeper understanding of trigonometry. It will also help with practising questions. It is advisable for students to solve questions that are related to triple-angle identities.

### Product identities

Students need to learn the product identities in order to solve the exercise questions. Revising the product identities is necessary to practice questions. All the difficult questions related to product identities can be practised well by taking help from NCERT solutions.

### Sum of Identities

The sum of identities is an important topic, and it is important for students to learn it. All of the questions framed by the “sum of identities” topic must be well practiced.

## Inverse Trigonometric Functions

Trigonometric Function Formulas do not technically have an inverse function because they are periodic and therefore not injective. Inverse trigonometric functions are multivalued functions because one can define an inverse function for every interval on which a Trigonometric Function Formulas is monotonic. In order to define a true inverse function, the domain must be constrained to an interval where the function is monotonic and is, consequently, bijective from this interval to the function’s image. Trigonometry, which is also a component of geometry, includes inverse Trigonometric Function Formulas. Formulas for Inverse Trigonometric Function Formulas help us understand how a right-angled triangle’s sides and angles relate to one another. Arcus functions and anti-trigonometric functions are other names for inverse Trigonometric Function Formulas in Mathematics. The inverse functions of the basic trigonometric function formulas sine, cosine, tangent, cosecant, secant, and cotangent are sine, cosine, tangent, cosecant, secant, and cotangent. Any trigonometric ratio can be used to find the angles using this method. In fields like Geometry, Engineering and others, inverse Trigonometric Function Formulas are frequently used.

## Trigonometric Functions Derivatives

formulas for the derivatives of the remaining four Trigonometric Function Formulas by first calculating the sine and cosine functions’ derivatives. Students can determine the velocity and acceleration of a simple harmonic motion by computing the derivatives of the sine and cosine functions.

## Integration of Trigonometric Function

Finding a function’s derivative, or rate of change, is the process of differentiation in Mathematics. The practical technique of differentiation, in contrast to the abstract nature of the theory that underlies it, can be performed by purely algebraic manipulations using three fundamental derivatives, four rules of operation, and a working familiarity with functions. A function that takes the anti-derivative of another function is known as an indefinite integral, though. An integral symbol (∫), a function, and a derivative of the function are used to represent it. An easier way to represent an anti-derivative is with the indefinite integral.

## Solved Examples on Trigonometric Functions

Regular question practise aids students in improving their exam preparation. It is important to be familiar with the curriculum. The outline of the topics and subtopics for the Mathematics exam is provided in the syllabus. Students are required to regularly practice questions from all of the chapters. For students to develop logical reasoning and analytical thinking abilities, Mathematics principles are crucial. It is crucial to practice Mathematics problems in primary classes because of this. Using the NCERT Solutions will assist students in developing a practise habit.

## Practice Questions on Trigonometric Functions

Sample papers and past years’ Mathematics papers are important in helping students improve their exam preparation. The marking scheme for Mathematics can be understood by looking at past years’ papers. Higher-weighted topics are supposed to be practised more frequently by students. Students’ level of preparation can be raised by solving papers from past years and sample papers. From Extramarks, they can download the most recent Mathematics curriculum. The syllabus aids students in developing a plan for effectively preparing for the Mathematics examination. Students’ confidence levels must be increased by answering chapter-related questions. Regular practice of exercises helps students become accustomed to the method of solving questions. Students are encouraged to solve as many questions related to Trigonometric Function Formulas as possible. Solving questions based on the Trigonometric Function Formulas will help students understand the topics in more depth. The Trigonometric Function Formulas are helpful in getting a thorough understanding of the concepts of trigonometry.