Sina = (opposite side of ∠ A)/(hypotenuse of ∠ A), COSA = (adjacent side of ∠ A)/(hypotenuse of ∠ A). One is Tan and the other is tg. Now we often use tan and tg as tangent function and ctg as cotangent function. In the new textbook, tan is used for tangent function and cot is used for cotangent function.

The relationship between the angles and sides of a triangle

Take the angle α as an example.

Sinα= opposite side: hypotenuse =BC:AC

Cosα= limb: hypotenuse =AB:AC

Tanα= opposite: limb =BC:AB

Cotα= edge: edge =AB:BC

Trigonometric function is a kind of transcendental function in elementary function in mathematics. Their essence is the mapping between the set of arbitrary angles and a set of ratio variables. The usual trigonometric function is defined in the plane rectangular coordinate system, and its domain is the whole real number domain. The other is defined in a right triangle, but it is incomplete.

Trigonometric function is one of the basic elementary functions, which takes the angle (the most commonly used radian system in mathematics, the same below) as the independent variable, and the angle corresponds to the coordinates where the terminal edge of any angle intersects with the unit circle or its ratio as the dependent variable. It can also be equivalently defined as the lengths of various line segments related to the unit circle.

The formulas of trigonometric functions seem to be many and complicated, but as long as we master the essence and internal laws of trigonometric functions, we will find that there is a strong connection between the formulas of trigonometric functions.

In trigonometric functions, there are some special angles, such as 30, 45 and 60. The trigonometric function values of these angles are simple monomials, and the specific values can be obtained directly in the calculation.

The following table shows the values of these functions:

The basis of learning trigonometric functions well is to understand the "unit circle". Definition: The unit circle is a circle with the origin as the center and the radius of 1 in the plane rectangular coordinate system.

Modern mathematics describes them as the limit of infinite sequence and the solution of differential equation, and extends their definitions to complex system.

It contains six basic functions: sine, cosine, tangent, cotangent, secant and cotangent. Because of the periodicity of trigonometric function, it does not have the inverse function in the sense of single-valued function.

Trigonometric functions have important applications in complex numbers. Trigonometric function is also a common tool in physics.

The acute trigonometric functions ABC, A, B and C in a right triangle are opposite sides of ∠A, ∠B and ∠C respectively, and ∠C is a right angle.

The following operating modes are defined:

Sin ∠A=∠A's opposite side length/hypotenuse length, and sin A is recorded as ∠A's sine; Sina = a/c cos ∠A = the length of the adjacent side/the length of the hypotenuse of ∠A, and cos A is recorded as the cosine of ∠ a; Cosa = b/c Tan ∠A = the opposite side length of ∠ A/the adjacent side length of ∠ A, Tana = Sina/Cosa = A/b Tan A is the tangent of ∠ A; When ∠A is an acute angle, sin A, cos A and tan A are collectively called "acute trigonometric function".

Sina = Cosb sinb = COSA Common trigonometric function In the plane rectangular coordinate system xOy, draw a ray OP from point O, let the rotation angle be θ, let OP=r, and the coordinate of point P be (x, y). ?

In this right triangle, Y is the opposite side of θ, X is the adjacent side of θ, and R is the hypotenuse, so the following six operation methods can be defined:

Basic function English expression language describes sine function sin sinθ= y/r angle θ. Are there more opposite sides than hypotenuse?

Cosine function cos θ=x/r angle θ, and the ratio of adjacent sides is hypotenuse?

Is the opposite side of tangent function tangent tan θ=y/x angle θ adjacent to the opposite side?

What are the adjacent sides of cotangent cot θ=x/y angle θ?

What is the hypotenuse of sec θ=r/x angle θ?

What is the hypotenuse of cotangent csc θ=r/y angle θ?

In junior and senior high school teaching, we mainly study three functions: sine, cosine and tangent. ?

Note: tan and cot used to be written as tg and ctg, but now they are not written like this.