# Definitional and value domains of trigonometric functions for any angle

### Definition and value domains of trigonometric functions

The definition domain of sin(x),cos(x) is R, and the value domain is [-1,1]; the definition domain of tan(x) is that x is not equal to π/2 + kπ, and the value domain is R; the definition domain of cot(x) is that x is not equal to kπ, and the value domain is R; and the value domain of y=a-sin(x)+b-cos(x)+c is [c- √(a&sup2;+b&sup2;),c+√(a&sup2;+b&sup2;)].

Definition

Trigonometric functions (also called “circular functions”) are functions of angles; they are important in the study of triangles and the modeling of periodic phenomena, as well as in many other applications. Trigonometric functions are usually defined as the ratio of the two sides of a right triangle containing the angle, or equivalently, as the lengths of various line segments on the unit circle. More modern definitions express them as solutions of infinite series or specific differential equations, allowing them to be extended to arbitrary positive and negative values, and even complex values.

Definitional and value domains

sin(x),cos(x) have a definitional domain of R and a value domain of [-1,1]

tan(x) has a definitional domain of x not equal to π/2 + kπ, and a value domain of R

cot(x) has a definitional domain of x not equal to kπ, and a value domain of R

y=a-sin(x) + b-cos(x) +c value domain is [c-√(a&sup2;+b&sup2;),c+√(a&sup2;+b&sup2;)]

Memorization mnemonic

Trigonometric function is a function of the quadrant sign coordinates note. Function image unit circle, period parity increase or decrease now.

The same angle relationship is very important, simplification and proof are needed. Hexagonal vertex, from top to bottom chord cut;

Center of the number one, link vertex triangle. Downward trigonometric sum of squares, inverse relationship is diagonal,

The vertex of any function, equal to the next two roots divided. Induced formula is good, negative to positive and then large to small,

To become acute angle is good to look up the table, the simplification of the proof of less. Half of two integer multiples, odd to even unchanged,

The latter will be regarded as an acute angle, the sign of the original function judgment. The cosine of the sum of two angles, into a single angle is good to find the value,

Cosine product minus the sine product, the angle of transformation of the many formulas. The sum and difference of the product must be the same name, the reciprocal angle change the name.

Calculate the proof of the angle first, pay attention to the structure of the function name, keep the basic amount of the same, difficult to easy to change.

The inverse principle as a guide, ascending power descending and difference product. The proof of the conditional equation, the idea of the equation shows the way.

The universal formula is not ordinary, into rational formulas first.

The formula is used smoothly and inversely, the use of deformation and clever use;

A plus cosine think cosine, a minus cosine think sine, the power to rise once the angle is halved, the power to rise and fall it for the model;

Trigonometric inverse function, the essence of the angle is to find the trigonometric function, the first trigonometric function, and then determine the range of the angle value;

Using the right-angled triangle, the image of a visual and good for a change of name, the equation of the triangle is simple, to the most simple solution set.

### What is the definition domain of trigonometric function?

The trigonometric function defining domains are: sin(x),cos(x) has a defining domain of R and a value range of [-1,1]; tan(x) has a defining domain of x not equal to π/2+kπ and a value range of R; cot(x) has a defining domain of x not equal to kπ and a value range of R; y=a-sin(x)+b-cos(x)+c has a value range of [c-√(a& amp;sup2;+b&sup2;),c+√(a&sup2;+b&sup2;)].

Memorization mnemonic

Trigonometry is a function, quadrant sign coordinates note. Function image unit circle, period parity increase or decrease now.

The same angle relationship is very important, simplification and proof are needed. At the vertex of the hexagon, the chord is cut from the top to the bottom.

Center the number one, connect the vertex triangles. Sum of squares of downward triangles, inverse relationship is diagonal.

Any function at the vertex is equal to the next two roots divided. Induction formula is good, negative to positive and then large to small.

To become an acute angle is good to look up the table, the simplification of the proof is not necessary. Half of two is an integer multiple, the odd number is converted to the even number and the even number remains unchanged.

The latter will be regarded as an acute angle, the sign of the original function judgment. The cosine of the sum of two angles is reduced to a single angle.

The product of cosines minus the product of sines is the formula for the transformation of angles. The product of sums and differences must have the same name, and the reciprocal angle changes its name.

Calculate the proof of the angle first, pay attention to the structure of the function name, keep the basic amount of the same, difficult to simple change.

Calculating the angle is the first step.

### What is the domain of definition of trigonometric functions?

The definition domain of sin(x) and cos(x) is R, and the value domain is [-1, 1]; the definition domain of tan(x) is x is not equal to π/2 + kπ, and the value domain is Rcot(x) is x is not equal to kπ, and the value domain is R.

The definition domain of a function refers to the range of values of a function in terms of its independent variable x. To find the definition domains of the trigonometric functions, you should be familiar with the various Trigonometric functions in each quadrant of the sign, and to note that the definition of the domain of each trigonometric function, generally expressed in radians.

Characteristics of trigonometric functions

Trigonometric functions, whether they are sine or cosine functions or tangent-coset functions, are all very regular functions, they are all periodic functions characterized by π, which determines a box characteristic of the independent variable of trigonometric functions, which can always be expressed in terms of π algebraically, and so π has become a core constant of the modified function. A core constant, with the periodic function values appear, there is also a regular jump in the graph, this jump is actually very beautiful, can be illustrated from the beauty of mathematical graphics and the beauty of complex mathematical research.

### What is the domain of definition of a trigonometric function?

sin(x),cos(x) has a definition domain of R and a value domain of [-1,1].

tan(x) is defined by x not equal to π/2+kπ, and has a value range of R.

cot(x) is defined by x not equal to kπ, and has a value range of R.

y=a-sin(x)+b-cos(x)+c has a value range of [c – √(a&sup2;+b&sup2;),c+√( a&sup2;+b&sup2;)].

Introduction

Trigonometry is one of the basic elementary functions, a function in which the angle (most commonly used in mathematics in radians, the same below) is the independent variable, and the angle corresponds to the coordinates of the intersection of the terminal side of an arbitrary angle with the unit circle, or the ratio thereof, is the dependent variable. It can also be equivalently defined in terms of the lengths of the various line segments associated with the unit circle.

Trigonometric functions play an important role in the study of the properties of geometric shapes such as triangles and circles, and are a fundamental mathematical tool in the study of periodic phenomena. In mathematical analysis, trigonometric functions are also defined as solutions of infinite series or specific differential equations, allowing their values to extend to any real or even complex value.

Common trigonometric functions include the sine, cosine, and tangent functions. Other trigonometric functions such as cotangent, tangent, cotangent, cosine, cosine, cosine, cosine, half-sine, half-cosine, and other trigonometric functions are used in other disciplines such as navigation, surveying and mapping, and engineering. The relationship between different trigonometric functions can be derived through geometric intuition or computation and is called the trigonometric constant.