# Proof using hyperbolic trig functions and complex variables

• Nerd2567
In summary, the conversation discusses a proof for the equation tan(2x) = -cos(b) / sinh(a) given x + yi = tan^-1 ((exp(a + bi)). The conversation includes the derivation of various equations and attempts to find a solution, but it is unclear if the solution is complete. Further clarification or additional theory may be needed to fully prove the equation.
Nerd2567
1. Given, x + yi = tan^-1 ((exp(a + bi)). Prove that tan(2x) = -cos(b) / sinh(a)

## Homework Equations

I have derived.
tan(x + yi) = i*tan(x)*tanh(y) / 1 - i*tan(x)*tanh(y)

tan(2x) = 2tanx / 1 - tan^2 (x)

Exp(a+bi) = exp(a) *(cos(b) + i*sin(b))[/B]3. My attempt:
By definition sinha = (exp(a) - exp (-a)) / 2

Let x + yi = z

tanz = exp(a + bi) = exp(a) * exp(bi)
hence; exp(a) = tan(z) / exp(bi)
exp(-a) = exp(bi) /tan(z)
therefore; exp(a) - exp(-a) = (tan(z) / exp(bi) ) * ( exp(bi) / tan(z) )
simplifying; exp(a) - exp(-a) = (tan^2(z) - exp(2bi)) / (exp(bi) * tan(z))
hence, sinh (a) = (exp(a) - exp(-a))/ 2 = (tan^2(z) - exp(2bi)) / 2(exp(bi) * tan(z))

Now am going to find -cos(b) --------

tanz = exp(a) * (cos(b) + isin(b))
cos(b) = (tan(z) / exp(a)) - isin(b)
-cos(b) = isin(b) - (tan(z) / exp(a))
simplifying;
-cos(b) = ( iexp(a)*sin(b) - tan(z) ) / exp(a)

therefore;
-cos(b) / sinh (a) =((( 2iexp(a + bi) * sin(b) tan(z) - 2exp(bi) * tan^2 (z) ))) / (( exp(a) tan^2(z) - exp(a +2bi) ))

this is pretty much my best attempt...:'( I must be missing theory please help...I will really appreciate it!

You can factor out several things which makes the expression shorter.
Also, the left side is real, so calculating the real part should be sufficient.

mfb said:
You can factor out several things which makes the expression shorter.
Also, the left side is real, so calculating the real part should be sufficient.
Perhaps, but I'd seem a bit illogical to do that since they said the right hand side should be tan(2x)...sighz the problem is either rather difficult or am missing a simple piece of theory to make it a 5 line proof (often the case) .

## 1. What are hyperbolic trig functions?

Hyperbolic trig functions are mathematical functions that are analogous to the traditional trigonometric functions (sine, cosine, tangent, etc.) but are based on the hyperbolic functions sinh (hyperbolic sine), cosh (hyperbolic cosine), and tanh (hyperbolic tangent). These functions are useful in many areas of mathematics, including complex analysis and differential equations.

## 2. How are hyperbolic trig functions used in proof using complex variables?

Hyperbolic trig functions are often used in conjunction with complex variables to prove mathematical theorems. They allow for the manipulation of complex numbers and can simplify complex expressions, making it easier to prove certain statements.

## 3. What is the relationship between hyperbolic trig functions and complex numbers?

Hyperbolic trig functions and complex numbers are closely related. They can be expressed in terms of each other, and many properties of hyperbolic trig functions can be derived from complex numbers. In particular, the complex exponential function can be used to express hyperbolic trig functions in terms of complex numbers.

## 4. Can hyperbolic trig functions be used in real-world applications?

Yes, hyperbolic trig functions have many real-world applications. They are often used in engineering, physics, and other fields to model and solve problems involving curved surfaces, such as the shape of a hanging chain or the trajectory of a projectile.

## 5. Are there any other commonly used trigonometric functions besides the hyperbolic ones?

Yes, in addition to the traditional trigonometric functions and the hyperbolic functions, there are also inverse trigonometric functions, such as arcsine, arccosine, and arctangent, as well as other hyperbolic functions like cosecant, secant, and cotangent. These functions have their own unique properties and applications.

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