Cotangent csc
(or cosec) tangential cots
(or ctg, ctn) [Edit] In addition to the six basic functions, there are four functions in history: secant and cotangent outside the normal vector [Edit] History With the realization that similar triangles keep the same proportion between their sides, there is an idea that there should be some standard correspondence between the sides of triangles and the angles of triangles. That is to say, for any similar triangle, for example, the ratio of the hypotenuse to the other two sides is the same. If the hypotenuse becomes twice as long, the other sides will become twice as long. These ratios are expressed by trigonometric functions. Hipachas in Nigeria (BC 180- 125), Ptolemy in Egypt (AD 90- 180), aryabhata (AD 476-550), Varahamihira, Brahmaputa, Hualazimi and so on all studied trigonometric functions. Nasir Ahldin Artusi, Giyas Al-Kahi (14+04th century), Ur Ugberg (14th century), Johannes Muller (1464), Retikus and his students Valentin Otto. Mad hava(c. 1400) of Sanggamagramma made an early study on the analysis of trigonometric functions in the form of infinite series. Euler's introduction in Infinite Analysis (1748) made the most important contribution to the establishment of the European trigonometric function. It also defines trigonometric function as infinite series, and expresses Euler formula and nearly modern abbreviations such as sin. Because. Don. ，cot。 , seconds. And cosec. [Edit] Definition of Right Triangle [Edit] There is only the definition of acute trigonometric function in right triangle. The sine of an acute angle is the ratio of its opposite side to its hypotenuse. SinA = opposite side/hypotenuse = a/h. The cosine of an acute angle is the ratio of its adjacent side to its hypotenuse. CosA= adjacent side/hypotenuse = b/h. The tangent of an acute angle is the ratio of its opposite side to its adjacent side. TanA = opposite side/adjacent side = a/b [Edit] In the rectangular coordinate system, let α be a quadrant angle in the plane rectangular coordinate system xOy, the last point of the angle, and the distance from P to the origin O, then the six trigonometric functions of α are defined as: function name defines function name defines sine cosine tangent cotangent [Edit] defines unit circle, and can also be defined according to unit circle with radius as the center. 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 provides a single visual image that encapsulates all the important trigonometric functions at once. According to Pythagoras theorem, the equation of unit circle is: in the image, given a common angle measured by radians. The counterclockwise measurement is a positive angle, and 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 cos θ and sin θ respectively. The triangle in this figure guarantees this formula; The radius is equal to the hypotenuse and the length is 1, so there are sin θ = y/ 1 and cos θ = x/ 1. The unit circle can be regarded as a way to view infinite triangles by changing the lengths of adjacent sides and opposite sides and keeping the hypotenuse equal to 1. Images of f(x) = sin(x) and f(x) = cos(x) functions on Cartesian plane. For greater than 2π or less than? 6? 12π angle, simply continue to rotate around the unit circle. 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 "primitive period" of this function. The basic period of sine, cosine, secant or cotangent is a whole circle, that is, 2π radians or 360 degrees; The basic period of tangent or cotangent is a semicircle, which is π radian or 180 degrees. Only sine and cosine are directly defined by the unit circle, and the other four trigonometric functions can be defined as: the image of the function f(x) = tan(x) on the Cartesian plane. 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. Instead, all basic trigonometric functions can be defined according to the unit circle with the center of O, similar to the geometric definition used in history. Especially for the chord AB of this circle, where θ is the diagonal half and sin(θ) is AC (half chord), which is the Indian definition of aryabhata (476-550 AD). Cos(θ) is the horizontal distance OC, and versin(θ) = 1? 6? 1 cos(θ) is a CD. Tan(θ) is the length of the tangent of line segment AE passing through A, so this function is called tangent. Cot(θ) is another tangent AF. Sec(θ) = OE and csc(θ) = OF are secant (intersecting the circle at two points) line segments, so they can be regarded as the projections of OA to the horizontal axis and the vertical axis along the tangent of A, respectively. Is DE exsec(θ) = sec(θ)? 6? 1 1 (the part that is cut into a circle). Through these constructions, it is easy to see that when θ approaches π/2 (90 degrees), secant function and tangent function diverge, while when θ approaches zero, cotangent function and cotangent function diverge. [Edit] The sine function defined by the series (blue) is an approximation of the 5th Taylor series (pink) of the whole circle with the center of the circle as the origin. Only by using the properties of geometry and limit, it can be proved that the derivative of sine is cosine and the derivative of cosine is negative sine. In calculus, all angles are measured in radians. Then Taylor series theory can be used to prove that the following identities are true for all real numbers X: These identities are often used to define sine and cosine functions. They are often used as the starting point for the serious processing and application of trigonometric functions (for example, in Fourier series), because the theory of infinite series is developed from the real number system and has nothing to do with any geometric considerations. The differentiability and continuity of these functions are often established separately from the definition of series itself. For other series, see: [1]}-Here is the N-degree up/down number, which is the N-degree Bernoulli number, and (below) is the N-degree Euler number. In this expression, the denominator is the corresponding factorial, while the numerator is called "tangent number", which has a combined explanation: they enumerate the alternating arrangement of finite odd potential sets. }-In this expression, the denominator is the corresponding factorial, and the numerator is called "secant number", which has a combinatorial explanation: they enumerate the interactive arrangement of finite even potential sets. From a theorem of complex analysis, this real function has a unique analytic extension to complex numbers. They have the same Taylor series, so the trigonometric functions defined on complex numbers use the Taylor series above. [Editor] The connection with exponential function and complex number can be proved from the above series definition that when the independent variable of complex exponential function is pure imaginary number, sine function and cosine function are both imaginary and real parts of complex exponential function. This connection was first noticed by Euler, and this identity is called Euler formula. In this way, trigonometric function becomes essential in geometric interpretation of complex analysis. For example, through the above identity, if we consider the unit circle on the complex plane defined by eix, as above, we can parameterize this circle according to cosine and sine, and the relationship between complex exponent and trigonometric function becomes very obvious. Furthermore, this allows to define the trigonometric function of the complex independent variable z: i2 =? 6? 1 1。 For pure real number x, it is also known that exponential processing is closely related to periodic behavior. [Editor] Definition of differential equation Both sine and cosine functions satisfy the differential equation, that is, each is a negative value of its second derivative. In the two-dimensional vector space V of all solutions of this equation, sine function is the only solution that satisfies the initial conditions y(0) = 0 and y ′ (0) =1,while cosine function is the only solution that satisfies the initial conditions y ′ (0) =1+0 and y ′ (0) = 0. Since sine and cosine functions are linearly independent, they together form the basis of V. This method of defining sine and cosine functions is essentially equivalent to using Euler formula. (See linear differential equation). Obviously, this differential equation can be used not only to define sine and cosine functions, but also to prove trigonometric identities of sine and cosine functions. In addition, it is observed that sine and cosine functions are satisfied, which means that they are characteristic functions of second-order operators. The tangent function is the only solution of the nonlinear differential equation satisfying the initial condition y(0) = 0. There is an interesting intuitive proof that the tangent function satisfies this differential equation; See Needham's visual complex analysis. [2][ Edit] The importance of radian radian specifies an angle by measuring the path length along the unit circle, which constitutes a specific radian angle of sine and cosine functions. In particular, only those sine and cosine functions that map radians into ratios can satisfy the classical differential equations that describe them. If the radian angle of sine and cosine functions is proportional to frequency, then the derivative is proportional to "amplitude". K here is a constant, indicating the mapping between units. If x is a degree, it means that the second derivative of sine of the degree does not satisfy the differential equation, but; The same is true of cosine. This means that these sine and cosine are different functions, so the fourth derivative of sine is sine, but its radiation angle is radian. [Editor] Trigonometric identities Main entry: Trigonometric identities have many identities in trigonometric functions. The most commonly used one is Pythagoras identity, which claims that the square of sine plus the square of cosine is always 1 for any angle. This can be obtained by Pythagorean theorem from right triangle with hypotenuse 1. In the form of symbols, Pythagoras' identity is: more often, it is written as a power of "2" after sine and cosine symbols; in some cases, the brackets can be omitted. Another key link is the formula of sum and difference. According to the sine and cosine of these two angles, the sine and cosine of the sum and difference of these two angles are given. They can be geometrically deduced by Ptolemy's argumentation method; It can also be obtained by Euler formula in algebra. When the two angles are the same, the summation formula is simplified to a simpler equation, which is called the double angle formula. These equations can also be used to derive the identity of product and difference, which was used in ancient times to convert the product of two numbers into the sum of two numbers and do faster operations like logarithm. Integral and derivative of trigonometric function can be found in derivative table, integral table and trigonometric function integral table. [Edit] Special values of trigonometric functions There are some commonly used special function values in trigonometric functions. The function name is 0sin0}-cos1}-tan01cot1sec1}-2csc2}-or ... (Of course, additional reduction is needed. ) function angle sincostan[ edit] inverse trigonometric function principal term: inverse trigonometric function trigonometric function is a periodic function, so it is not an injective function, so strictly speaking, there is no inverse function. Therefore, in order to define inverse functions, we must define their domains so that trigonometric functions are bijective functions. The function on the left below is defined by the equation on the right; These don't prove identity. The basic inverse function is usually defined as: for the inverse trigonometric function, the symbol sin? 6? 1 1 and cos? 6? 1 1 is often used in arcsin and arccos. When using this symbol, the inverse function may be confused with the reciprocal of this function. Using symbols prefixed with "arc-" can avoid this confusion, although "arcsec" may occasionally be confused with "arcsecond". Just like sine and cosine, the inverse trigonometric function is defined in terms of infinite series. For example, these functions can also be defined by proving that they are indefinite integrals of other functions. For example, sine function can be written as the following integral: a similar formula can be found in the entry of inverse trigonometric function. Using complex logarithm, these functions can be extended to complex radial angles:
[Editor] As the name implies, the properties and applications of trigonometric functions are very important in trigonometry, mainly because of the following two results. [Editor] Sine Law Sine law claims that for any triangle whose sides are A, B and C and the angles relative to these sides are A, B and C, there are: also expressed as: Lissajous curve, an image formed by the function of the triangle bottom. Divide a triangle into two right-angled triangles, which can be proved by the definition of sine above. Sina/A in this theorem is the reciprocal of the diameter of a circle passing through A, B and C. Sine theorem is used to calculate the length of an unknown side when two angles and one side of a triangle are known. This is a common situation in triangulation. [Editor] Cosine law Cosine law (also called cosine formula) is a generalization of Ptolemy's theorem, which can also be proved by dividing a triangle into two right-angled triangles. Cosine law is used to determine unknown data when two sides and an angle of a triangle are known. If the angle is not included between these two sides, the triangle may not be unique (edge-edge-angle congruence ambiguity). Be careful of this ambiguity of cosine law. [Edit] Other useful attributes include the tangent law: [Edit] The additive analysis of animation, the periodic function of square wave and increasing harmonic number. Trigonometric functions are also very important in physics. For example, sine and cosine functions are used to describe simple harmonic motion, which simulates many natural phenomena, such as the vibration of the weight attached to the spring and the small-angle swing of the weight hanging on the rope. Sine and cosine functions are one-dimensional projections of circular motion. Trigonometric functions have also been proved to be very useful in the study of general periodic functions. These functions have characteristic waveforms as images, which are useful for simulating cyclic phenomena such as sound waves or light waves. All signals can be written as the sum of sine and cosine functions of different frequencies (usually infinite); This is the basic idea of Fourier analysis, which uses trigonometric series to solve boundary value problems of various differential equations. For example, a square wave can be written as a Fourier series. In the animation on the right, you can see that a very good approximation is generated with only a few items.