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In Mathematics, Hyperbolic functions refer to the exponential functions that share similar properties to trigonometric functions. Such hyperbolic functions are analogous trigonometric functions in that they are named the same as trigonometric functions with the letter ‘h’ appended to each name. The hyperbolic function formula has the same relationship to the hyperbola that trigonometric functions have to the circle. This is why they are collectively known as hyperbolic functions and are individually called hyperbolic sine, hyperbolic cosine, and so on. Hyperbolic functions can be used as solutions to some types of partial differential equations. Here we will discuss the hyperbolic functions formula, general equation of hyperbola, standard equation of hyperbola, hyperbola formula, trigonometric hyperbolic formulas.
| Table of Content |
Key Takeaways: Exponential functions, Trigonometric functions, Sine, Cosine, Tangent, Hyperbola, Hyperbolic function, Differential equation, hyperbolic sine, hyperbolic cosine
Definition of Hyperbola
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A hyperbola refers to the plane curve that is generated by a point so moving that the difference of the distances from two fixed points is constant. The foci refer to the two fixed points that are the mid-point of the line segment joining the foci is the center of the hyperbola. The line through the foci is known as the transverse axis. The conjugate axis refers to the line perpendicular to the transverse axis and through the. The points at which the hyperbola intersects the transverse axis are the vertices of the hyperbola. The distance between the two foci is given by 2c and the length of the conjugate axis is given by 2b. And the distance between the two vertices is 2a. The length of the transverse axis is 2a s. The formula for b is sqrt(c2–a2).
- Hyperbolic sine of x: sinh x = (ex – e−x) / 2
- Hyperbolic cosine of x: cosh x = (ex – e−x) / 2
- Hyperbolic tangent of x: tanh x = (ex – e−x) / (ex – e−x)
- Hyperbolic cotangent of x: coth x = (ex – e−x) / (ex – e−x)
- Hyperbolic secant of x: sech x = 2 / (ex – e−x)
- Hyperbolic cosecant of x: csch x = 2 / (ex – e−x)
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| Quadrilateral Formula | Trapezoid Formula | Isosceles Triangle Theorems |
Relationship Among Hyperbolic Functions and Derivatives
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- Tanh x = (sinh / x) / cosh x
- Coth x = 1 / tanh x = cosh x / sinh x
- Sech x = 1 / cosh x
- Csch x = 1 / sinh x
- Cosh2 x – sinh2 x = 1
- Sech2 x + tanh2 x = 1
- Coth2 x – csch2 x = 1
- d / dx sinh x = cosh x
- d / dx cosh x = sinh x
- d / dx tanh x = sech2 x
- d / dx sech x = −sech x × tanh x
- d / dx coth x = −csc h2 x
- d / dx csch x = –csch x × coth x
Things to Remember
- Hyperbolic functions refer to the exponential functions that share similar properties to trigonometric functions.
- The distance from the fixed point in the plane has a constant ratio that is greater than the distance from the fixed point in a plane.
- sinh (x+y) =sinh x cosh y + cosh x sinh y
- cosh (x+y) = cosh x cosh y + sinh x sinh y
- tanh (x+y) = (tanh x + tanh y) / 1+tanh x. tanh y
- tanh (x-y) = (tanh x - tanh y) / 1-tanh x. tanh y
- coth (x+y) = (coth x. coth y+1) / coth y. coth x
- coth (x-y) = (coth x. coth y-1) / coth y. coth x
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Sample Questions
Ques. What are Hyperbolic Functions? (2 marks)
Ans. We have studied trigonometric functions which are defined on or for a circle in a similar way to define hyperbola we use the hyperbolic function. Generally, we use sine, cosine and other functions in trigonometry. In a similar manner, we use csc h, sec h, sin h, tan h, cot h, and cos h. For trigonometry, we know that the points of coordinates on the circle unit are (sin Φ, cos Φ). In a similar way (sin Φ, cos Φ) forms the right half of the equilateral hyperbole for hyperbolic functions.
Ques. What are Inverse Hyperbolic Functions? (2 marks)
Ans. As we have studied, the inverse of the hyperbola is known as the inverse of hyperbolic functions. For an instance, if y = sinh^-1 x, then x = sinh y is the inverse of the sine of hyperbolic functions. The inverse of the hyperbolic functions is expressed into the terms of function logarithm. Similar to the inverse of the hyperbole being used in another way on the contrary trigonometric functions are useful in certain integration, calculus. Few restrictions follow on the domain to make functions into one to one of each and the domains resulting and inverse functions of their ranges.
Ques. Explain are Cosh x and Sinh x. (2 marks)
Ans In mathematics, hyperbolic functions can generally be defined as analogs of the trigonometric functions in mathematics that are defined for the hyperbola rather than on the circle (unit circle). Just as the points (cos t, sin t) and we use a circle with a unit radius, the points generally (cosh t, sinh t) form the right half of the equilateral hyperbola. The above mentioned are the functions used for disclosing the right angle shapes. Cos x and Sin x are trigonometric identities. Sin x represents the proportion diametric to the hypotenuse. Cos represents the proportion adjacent to the hypotenuse.
Ques. What are the six trigonometric formulas? (2 marks)
Ans. Sine, cosine, secant, cosecant, tangent, and cotangent are the six trigonometric functions. A right-angled triangle is used as a reference for this, the trigonometric functions or identities that are derived: sin θ = Opposite Side/Hypotenuse. sec θ = Hypotenuse/Adjacent Side. cos θ = Adjacent Side/Hypotenuse. They are used in geometry-related science for example celestial mechanics, navigation, and many more. Angle functions, circular functions and goniometric functions are other names for such trigonometric instruments.
Ques. Find d / dt cothe2tx. (2 marks)
Ans. d / dt cothe2tx
= (– csc h2 e2tx) e2tx (2x)
= –2 xe2tx / csch2e2tx
Ques. State the uses of Hyperbolic functions? (2 marks)
Ans. The hyperbolic cosine function can be used to describe the shape of the curve formed by a high-voltage line suspended between two towers (see catenary). A measure of distance in certain types or kinds of non-Euclidean geometry can be defined using Hyperbolic functions. Hyperbolic functions have many applications in science, math, physics, and engineering. Such functions came from the imaginary parts of cos and sin from the whole complex plane. ‘Nikolay Ivanovich Lobachevsky’, a Russian mathematician had discovered Hyperbolic functions.
Ques. State the 6 hyperbolic functions. (3 marks)
Ans. Following are the 6 hyperbolic functions:
- Sinh x or hyperbolic sine
- Cosech x or hyperbolic cosecant
- Cosh X or hyperbolic cosine
- Tanh x or, hyperbolic tangent
- Coth x or hyperbolic cotangent
- Sech x or hyperbolic secant
Ques. What are SOH, COH, and TOA? (3 marks)
Ans. It is a mnemonic way or trick to remember the three basic trigonometric ratios defined by the trigonometric ratio definition.
SOH represents Sine equals Opposite over Hypotenuse. (Sin (θ) = Opposite/Hypotenuse)
CAH represents Cosine equals Adjacent over Hypotenuse. (Cos (θ) = Adjacent/Hypotenuse)
And, TOA represents Tangent equals Opposite over Adjacent. (Tan (θ) = Opposite/Adjacent)
We easily remember the functions of trigonometry using SOH, CAH, TOA abbreviations, it is the most vital function that helps us in many ways.
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