Definition of trigonometric function of an arbitrary angle

Definition of trigonometric function of an arbitrary angle: let a be an arbitrary angle whose terminal side intersects the unit circle at the point P(x, y), then sina=y, cosa=x,tana=(x≠0).

Geometric representation: Trigonometric lines can be seen as geometric representations of trigonometric functions, with sine lines all starting on the x-axis, cosine lines all starting at the origin, and tangent lines all starting at (1, 0). The oriented lines MP, OM, and AT are called the sine, cosine, and tangent of angle a, respectively, as shown in the figure.

Related concepts of angles:

1. From the point of view of motion, angles can be categorized into positive, negative and zero angles.

2. From the point of view of the position of the final side, angles can be divided into quadrant angles and axial angles.

3. If B and a are angles with the same final side, then B is expressed in terms of a as B = 2kit + a, kZ.

In the right-angle coordinate system, the radius of a circle is 1. The trigonometric function of an arbitrary angle α is defined as follows:

Sine: The ratio of the longitudinal coordinates of the point of intersection A of ∠α with the unit circle to the circle’s radius is called the sine, and it is expressed as: sinα=Ay /OA=Ay; where Ay is called the sine.

Cosine: The ratio of the horizontal coordinate of the intersection of ∠α with the unit circle, A, to the radius of the circle is called the cosine, and is expressed as: cosα=Ax/OA=Ax; where Ax is called the cosine.

Tangent: the ratio of the vertical coordinate to the horizontal coordinate of the intersection A of ∠α and the unit circle is called the tangent, expressed as: tanα=Ay/Ax.

In any angle triangle, the side angles have the following functional relationships:

1. The sine theorem: in an arbitrary angle triangle, the sines of the individual angles are equal to the ratios of the sides to which they are opposite and equal to the the diameter of the outer circle.

2. Cosine Theorem: In any angular triangle, the square of any side is equal to the product of the sum of the squares of the remaining two sides minus twice the product of these two sides and the cosine of their angle.

### Definition of Trigonometric Functions of Arbitrary Angles

Trigonometric functions are a class of functions in mathematics belonging to the transcendental functions of elementary functions. They are essentially mappings between a set of arbitrary angles and a set of variables of a ratio. The usual trigonometric functions are defined in the plane right-angle coordinate system, and their domain of definition is the entire domain of real numbers. Another definition is in right triangles, but not exactly. Modern mathematics describes them as limits of infinite series and solutions of differential equations, extending their definition to the complex number system. It contains six basic functions: sine, cosine, tangent, cotangent, secant, and cosecant. Because of their periodicity, trigonometric functions do not have inverse functions in the sense of single-valued functions. Trigonometric functions have more important applications in complex numbers. Trigonometric functions are also commonly used in physics.

In the plane right-angle coordinate system xOy, a ray OP is drawn from the point O. Let the angle of rotation be θ, let OP=r, and the coordinates of the point P be (x,y).

In this right triangle, y is the opposite side of θ, x is the neighbor of θ, r is the hypotenuse, then the following six operations can be defined:

Basic FunctionsEnglish ExpressionsLanguage DescriptionSine FunctionSinesinθ=y/rAngle α of the opposite side of the hypotenuseCosine FunctionCosinecosθ=x/rAngle α of the neighbor of the hypotenuseTangent Function Tangenttanθ = y/x angle α’s contralateral-ratio adjacency cotangent function Cotangentcotθ = x/y angle α’s adjacency-ratio contralateral secantsecθ = r/x angle α’s skew-ratio adjacency cosecantcscθ = r/y angle α’s skew-ratio contralateral Note: tan, cot used to be written as tg, ctg, but are no longer used in This writing style. Very common trigonometric functions

In addition to the above six common functions, there are some uncommon trigonometric functions, these operations have tended to be eliminated:

Function name and common function transformation relationship positive vector function versinθ=1-cosθ cosinθ cosinθ coversθ=1-sinθ half-posinθ function haversθ=(1-cosθ)/2 half cosinθ function hacoversθ=(1-sinθ)/2 external positive cut function exsecθ=secθ-1 external cosine cut function excscθ=cscθ-1