Trigonometric function is a transcendental function in elementary functions in mathematics. Their essence is the mapping between any set of angles and a set of ratio variables. The usual trigonometric function is defined in a plane rectangular coordinate system. Its definition field is the whole real number field.

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. And mastering the inherent law and essence of trigonometric function is also the key to learn trigonometric function well. The following are ways to show these internal laws and connections through mind mapping, which is convenient for learners to master trigonometric functions. Figure 1 shows the main branches of learning trigonometric functions. Let's start with the branches below, one by one.

Two: Angle and curvature system

2. 1 As we know, there are two commonly used measurement methods: angle system and arc system. What is an angle system? The so-called angular system is to divide the circumference into 360 equal parts, in which the central angle corresponding to 1 equal part is defined as 1 degree and recorded as 1 degree. And 1/60 of 1 degree is defined as1point and recorded as1'; 1/60 of 1 is defined as 1 sec and recorded as1". In other words, 1 = 60', 1'= 60 ". Fig. 2 is a schematic diagram of an angle system.

2.2 According to the quantitative relationship among central angle, arc length and radius, the radian system is introduced. When the arc length is equal to the radius, the central angle corresponding to the arc is 1 radian, which is recorded as 1rad. Positive angle radians are positive, negative angle radians are negative, and zero angle radians are negative. If the radian of the central angle α of a circle with radius r is l, then the absolute value of the radian number of the angle α is | α| = l/r.

2.3 Angle system and circular arc system conversion, digital representation and graphic representation are as follows.

2.3. 1 Digital expressions of angle system and arc system:

360 = 2π radians

180 = π radian

1 =(π/180) rad ≈ 0.0 1745 rad.

1 radian =( 180/π) ≈57.30

The angle of alpha degree =? α? (π/180) radian

2.3.2 Angle system and arc system are shown in Figure 3:

2.4 Figure 4 is the mind map of angle system and arc system.

Third, the basic properties of trigonometric functions

3. Definition of1trigonometric function. In a right triangle, when the connecting lines AB, AC and BC of three points A, B and C on the plane form a right triangle, where ∠ACB is a right angle. For ∠BAC, if the opposite side a = BC, the hypotenuse) c=AB, and the adjacent side b = AC, there exists as shown in Figure 5:

3.2 The sign of trigonometric function is determined by its quadrant. As shown in figs. 6 and 7.