Definition of trigonometric function

The acute trigonometric functions (3 blocks) in right triangle ABC, A, B and C are the opposite sides of ∠A, ∠B and ∠C respectively, and ∠C is a right angle. Then the following operation modes are defined: sin ∠A=∠A = the length of the opposite side/the length of the hypotenuse of ∠A, and sin A is recorded as the sine of ∠ a; 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 functions

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: English expression language description of basic functions.

Sin θ=y/r angle θ, and the opposite side is greater than the hypotenuse.

Cosine function cos θ=x/r The adjacent side of angle θ is larger than the hypotenuse.

The opposite side of tangent function tangent tan θ=y/x angle θ is adjacent to the opposite side.

Cotangent cot θ=x/y angle θ compares the adjacent edges of the edge.

The hypotenuse sec θ=r/x angle θ of the secant is adjacent to the edge.

The hypotenuse θ of cotangent csc = R/Y angle θ comparison edge.

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. sinπ/3

Unconventional trigonometric function

In addition to the above six commonly used functions, there are some trigonometric functions that are not commonly used and tend to be eliminated: the transformation relationship between function names and commonly used functions.

Positive vector function version θ= 1-cosθ

Covector function coversθ= 1-sinθ.

Semipositive vector function haversθ = (1-cos θ)/2;

Semi-cofactor function HaCoversθ = (1-sinθ)/2;

Exosecθ = secθ- 1。

Cotangent function exccscθ = cscθ-1.

Definition of unit circle

You can also define six trigonometric functions according to the unit circle with the radius of 1 and the center of the circle as the origin. The definition of unit circle is of little value in practical calculation. In fact, for most angles, it depends on the right triangle. But the definition of the unit circle does allow trigonometric functions to define all positive and negative radians, not just the angle between 0 and π/2 radians. It also provides images containing all the important trigonometric functions. According to Pythagorean theorem, trigonometric function

The equation of the unit circle is: x 2+y 2 =1Some commonly used angles measured in radians are given in the image. The counterclockwise measurement is a positive angle, while the clockwise measurement is a negative angle. Let a straight line passing through the origin make an angle θ with the positive half of the X axis and intersect the unit circle. The x and y coordinates of this intersection point are equal to cosθ and sinθ respectively. The triangle in the image ensures this formula; The radius is equal to the hypotenuse and the length is 1, so there are sinθ = y/ 1 and cosθ = x/ 1. The unit circle can be regarded as a way to view an infinite number of triangles by changing the lengths of adjacent sides and opposite sides, but keeping the hypotenuse equal to 1. For angles greater than 2π or less than or equal to 2π, you can continue to rotate around the unit circle directly. In this way, sine and cosine become periodic functions with a period of 2π: for any angle θ and any integer k, the minimum positive period of the periodic function is called the "basic period" of this function. The basic period of sine, cosine, secant or cotangent is a complete circle, that is, 2π radians or 360; The basic period of tangent or cotangent is a semicircle, which is π radian or 180. Only sine and cosine are directly defined by the unit circle, and the other four trigonometric functions are defined as shown in the figure. The definition of the other four trigonometric functions

In the image of tangent function, it changes slowly around the angle kπ, but changes rapidly at approach angle (k+ 1/2)π. The image of tangent function has a vertical asymptote at θ = (k+ 1/2)π. This is because when θ connects to (k+ 1/2)π from the left, the function approaches positive infinity, and when θ approaches (k+ 1/2)π from the right, the function approaches negative infinity. On the other hand, all basic trigonometric functions can be defined according to the unit circle whose center is O, similar to the geometric definition used in history. Special trigonometric function

But for the chord AB of this circle, where θ is half of the diagonal and sin θ is AC (half chord), this is the definition of India's Ayabata intervention. Cosθ is the horizontal distance OC, and versin θ = 1-cosθ is CD. Tanθ is the length of the tangent of line segment AE through A, so this function is called tangent. Cotθ is another tangent AF. Secθ =OE and csθ= OF are secant (intersecting the circle at two points) line segments, so they can be regarded as the projections of OA along the tangent of A to the horizontal axis and vertical axis respectively. DE is exsecθ = secθ- 1 (the part cut out of the circle). Through these constructions, it is easy to see that when θ approaches π/2, secant function and tangent function diverge, while when θ approaches zero, cotangent function and cotangent function diverge.

trigonometric function line

According to the definition of the unit circle, we can make three directed line segments (vectors) to represent the values of sine, cosine and tangent. As shown in the figure, the circle O is the unit circle, P is the intersection of the terminal edge of α and the unit circle, M is the projection of P on the X axis, S( 1, 0) is the intersection of the circle O and the positive semi-axis of the X axis, and the tangent L of the circle O is made through the S point. Then the vector MP corresponds to the sine value of α and the vector OM corresponds to the cosine value. The intersection of the extension line (or anti-extension line) of OP and L is T, so the vector s t corresponds to the tangent value. The starting point and ending point of a vector cannot be reversed, because its direction is meaningful. With the help of line trigonometric function line, we can observe that the sine value of the second quadrant angle α is positive, the cosine value is negative and the tangent value is negative. 1, the acute trigonometric function defines the sine (sin), cosine (cos), tangent (tan), cotangent (cot) and secant (sec) of acute angle A, and cotangent csc is called the acute trigonometric function of angle A, and sine is equal to the hypotenuse of the opposite side; Cosine (cos) is equal to the ratio of adjacent side to hypotenuse;

The tangent (tan) of trigonometric function (8 sheets) is equal to the opposite side; Cotangent is equal to the comparison of adjacent edges; Secant is equal to the hypotenuse than the adjacent edge; Cotangent (csc) is equal to the ratio of hypotenuse to edge. 2. The relationship of trigonometric functions with complementary angles: SIN (90-α) = COS α, COS (90-α) = SIN α, TAN (90-α) = COT α, and Cot (90-α) = tan α. 3. Relationship quotient between trigonometric functions with the same angle: sina/COSA = tana. Square relation: sin 2 (a)+cos 2 (a). The relation of = 1 product: sina = tanacosa cosa = cota sina cota = cosacscatana cota =1reciprocal relation: in the right triangle ABC, the sine value of angle A is equal to the opposite side of angle A, and the cosine is equal to the tangent of the adjacent side of angle A, which is equal to the opposite side. Cotangent is equal to trigonometric function value 4, trigonometric function value (1) and trigonometric function value (2) of special angle compared with adjacent edges at any angle between 0 and 90. Look up the trigonometric function table (3) Changes of trigonometric functions of acute angles (i) All trigonometric functions of acute angles are positive values (ii) When the angle changes from 0 to 90, the sine value increases (or decreases), the cosine value decreases (or increases), and the tangent value increases (or decreases). Tana >: 0, cotA & gt0. The special trigonometric function value is 0 30 45 60 9001/2 √ 2/2 √ 3/21√ Sina1√ 3/2 √ 2/kloc-0. 31√ 3none √ tananone √ 31√ 3/30 √ cota "acute trigonometric function" belongs to trigonometry and is an important content in the field of "space and graphics" of mathematics curriculum standard. According to the mathematics curriculum standard, middle school mathematics divides trigonometry into two parts, the first part is in the third stage of compulsory education, and the second part is in the high school stage. The third stage of compulsory education mainly studies the content of acute triangle function and the solution of right triangle. This textbook has a chapter called "Acute Trigonometric Function". Trigonometry in senior high school is the main part of trigonometry, including solving oblique triangles, trigonometric functions, inverse trigonometric functions and simple trigonometric equations. The former part is the important foundation of the latter part, both in content and way of thinking. Mastering the concept of acute trigonometric function and the method of solving right triangle is an important preparation for learning trigonometric function and solving oblique triangle.