With regard to hyperbolic functions, find the inverse function of hyperbolic functions

What is the curve of the naturally sagging chain shape?

Not hyperbola anyway.

Ha ha. However, hyperbola does have a wide range of applications.

Under the action of the inverse square of the square.

And other things related to conic curves.

y=ln(x+ (y +1)) is the inverse function of the hyperbolic sinusoid. The hyperbolic sine function is a type of hyperbolic function, which is generally written as sinh in mathematical language, and can also be abbreviated as sh.

Like trigonometric functions, hyperbolic functions are also divided into 6 types: hyperbolic sine, hyperbolic cosine, hyperbolic tangent, hyperbolic cotangent, hyperbolic secant, hyperbolic cosecant, hyperbolic cosecant, hyperbolic cosine function and hyperbolic cosine function are the two most basic hyperbolic functions, from which the hyperbolic tangent function can be deduced and so on.

Function properties: y=sinh x, definition domain: r, value range: r, odd function, the function image is a strictly monotonically increasing curve that crosses the origin and crosses the , quadrant, and the function image is symmetrical with respect to the origin.

y=cosh x, definition domain: r, value range: [1,+ even function, the function image is a catenary, the lowest point is (0,1), in the quadrant part is a strictly monotonic increasing curve, and the function image is symmetric with respect to the y-axis.

y=tanh x, definition domain: r, value range: (-1,1), odd function, the function image is a strict monotonically increasing curve that crosses the origin and crosses the quadrant, and its image is confined to two horizontal asymptotic lines y=1 and y=-1.

y=shx=1/2(e^x-e^(-x))2(e^x)*y=e^(2x)-1

e^(2x)-2y(e^x)-1

e x=1 2*(2y+ (4y +4)) takes the positive sign, the negative sign is meaningless) = y+(y +1) (1 2).

x=ln(y+√(y²+1))

Or written as y=ln(x+ (y +1)) is a hyperbolic sinusoidal inverse function, hyperbolic cosine inverse function, similar derivation.

y=ln(x+√(y²-1))

Arch should be discussed in two parts, the inverse function of coshx in the region less than 0 should be ln (y-root y-square minus one).

shx hyperbolic sinusoidal function.

A hyperbolic sine function is a type of hyperbolic function. The hyperbolic sine function is generally written as sinh in mathematical language, and can also be abbreviated as sh. Like trigonometric functions, hyperbolic functions are also divided into 6 types: hyperbolic sine, hyperbolic cosine, hyperbolic tangent, hyperbolic cotangent, hyperbolic secant, hyperbolic cosecant, hyperbolic cosecant, hyperbolic cosine function and hyperbolic cosine function are the two most basic hyperbolic functions, from which the hyperbolic tangent function can be deduced and so on.

The hyperbolic sine function is defined as: sinh=[e x-e (-x)] 2.

In mathematics, hyperbolic functions are a class of functions that are similar to common trigonometric functions, also called circular functions. The most basic hyperbolic functions are the hyperbolic sine function sinh and the hyperbolic cosine function leakage cavity number cosh, from which the hyperbolic tangent function tanh and so on can be derived, and its derivation is also similar to the derivation of trigonometric functions. The inverse function of the hyperbolic function is called the inverse hyperbolic function.

The domain of the hyperbolic function is the interval, and the value of its independent variable is called the hyperbolic angle. Hyperbolic functions appear in the solution of some important linear differential equations, such as the definition of catenary lines and the Laplace equation.

The hyperbolic function can be defined with the help of an exponential function.

sinh_cosh_tanh

The hyperbolic sine is vertical.

sh z =(e^z-e^(-z))/2

Hyperbolic cosine. ch z =(e^z+e^(-z))/2

Hyperbolic tangent. th z = sh z /ch z =(e^z-e^(-z))/e^z+e^(-z))

Hyperbolic cotangent. cth z = ch z/sh z=(e^z+e^(-z))/e^z-e^(-z))

Hyperbolic secant. sch z =1/ch z

Hyperbolic cosecant. xh(z) =1/sh z

The origin of hyperbolic functions is catenary, and the first person to raise the problem of catenary shape was Leonardo da Vinci. He pondered the shape of the black necklace around the woman's neck when he painted "The Woman Holding the Silver Sable", but unfortunately he died without receiving an answer.

After 170 years, the famous Jacob Bernoulli raised this question again in an article and tried to prove that it was a parabola. In fact, both Galileo and Girard, before Jacob Bernoulli, speculated that the curve of the chain was parabolic.

Within the real domain, the value of a trigonometric function is defined by the length of the trigonometric line on the unit circle and the end edge of the corner. Of course, there are positive and negative aspects of this length. In the same way, the value of the hyperbolic function is also defined by the length of the hyperbolic line on the hyperbolic and the end edge of the corner.

In mathematics, hyperbolic functions are similar to common trigonometric functions (also called circular functions). The basic hyperbolic functions are hyperbolic sine "sinh", hyperbolic cosine "cosh", from which hyperbolic tangent "tanh" is derived, etc. It is also similar to the derivation of trigonometric functions.

The inverse function is the inverse hyperbolic sine "arsinh" (also called "arcsinh" or "asinh") and so on.

If you are a high school student, you need to know what his function image looks like, whether it is an odd function or an even function, and you don't need to know the series and some complex functions.

Search: The origin of hyperbolic functions (including hyperbolic sine, hyperbolic cosine).

Summary. What are the hyperbolic functions.

A common type of hyperbolic function.

What is the conclusion drawn in the oblique coordinate system?

Hyperbolic functions are a class of functions related to hyperbola. In mathematics, hyperbola is a class of quadratic curves defined as all points on a plane such that the difference in distance from that point to two given points (called the focal point) is equal to the absolute line force of one constant (called the eccentricity of the hyperbola).

Hyperbolic functions are functions determined by x and y based on the properties of hyperbolic bridges. Common hyperbolic functions include hyperbolic sine function, hyperbolic cosine function, hyperbolic tangent function, and hyperbolic cotangent function. They are defined as follows:

Hyperbolic sinusoidal function sinh(x) =e x - e -x) 2

Hyperbolic cosine function cosh(x) =e x + e -x) 2

The hyperbolic tangent function tanh(x) =sinh(x) cosh(x) =e x - e -x) e x + e -x).

hyperbolic cotangent function coth(x) =cosh(x) sinh(x) =e x + e -x) e x - e -x).

Hyperbolic functions have a wide range of applications in mathematics, such as in calculus, number theory, physics, and engineering. They are also used to solve different types of calculus equations, data analysis, and image processing.

The opposing power refers to the inverse trigonometric function, the logarithmic function, the power function, the trigonometric function, and the exponential function. The order of partial points is considered from back to front. This is just an abbreviation for the simple usage when using the partial integration method.

The main principle of the partial integration method is to use the differential formula of two multiplicative functions to convert the required integration into the integration of another relatively simple function. Wheel pins.

f1(-c,0), f2(c,0) are hyperbolic c:

x 2 a 2-y 2 b 2 = 1 (jujube lift closed a 0, b 0, c 2 = a 2 b 2) to obtain 2 foci.

p(x0,y0) is a point on c, we call pf1 and pf2 are the stool coke radius of the double line, then pf1 = a ex0), pf2 = ex0-a), (e=c a is the eccentricity) take " when the point is on the right branch of the hyperbola " and take "-" when the point is on the left branch of the hyperbola

In a planar Cartesian coordinate system, the image of the binary quadratic equation h(x,y)=ax 2+bxy+cy 2+dx+ey+f=0 satisfies the following conditions.

1.a, b, and c are not all 0s.

2. b^2 - 4ac > 0。

x^2/a^2 - y^2/b^2 = 1。