# Shine Bright: Solving the Equation \sinh 3x = 4\sinh^3x + 3 \sinh x

• chwala
In summary: This is not a proof. It is better to use the properties of sinh and cosh directly, as suggested by the OP.The first equality is incorrect. The second equality is correct. The third equality is incorrect. The fourth equality is correct. The fifth equality is incorrect.In summary, using the properties of hyperbolic trigonometric functions, we can prove that ##\sinh 3x = 3\sinh x+ 4\sinh^3x##.
chwala
Gold Member
Homework Statement
Prove the hyperbolic function corresponding to the given trigonometric function.

##\sin 3x = 3\sin x- 4\sin^3x##
Relevant Equations
hyperbolic trig. properties.
We shall have;

##\sinh 3x = 3\sinh x- 4\sinh^3x##

##\sinh 3x =\sinh (2x+x)=\sinh 2x \cosh x + \cosh 2x \sinh x##

##\sinh 3x= 2\sinh x \cosh x \cosh x + (1+2 \sin^2x) \sinh x##

##\sinh 3x=2 \sinh x \cosh^2 x + \sinh x + 2\sinh^3x##

##\sinh 3x= 2\sinh x + 2\sinh^3 x + \sinh x + 2\sinh^3x##

##\sinh 3x = 4\sinh^3x + 3 \sinh x##

Insight welcome...today i have been singing shine shine shine

chwala
chwala said:
Homework Statement:: Prove the hyperbolic function corresponding to the given trigonometric function.

##\sin 3x = 3\sin x- 4\sin^3x##
Relevant Equations:: hyperbolic trig. properties.

We shall have;

##\sinh 3x = 3\sinh x- 4\sinh^3x##

##\sinh 3x =\sinh (2x+x)=\sinh 2x \cosh x + \cosh 2x \sinh x##

##\sinh 3x= 2\sinh x \cosh x \cosh x + (1+2 \sin^2x) \sinh x##

##\sinh 3x=2 \sinh x \cosh^2 x + \sinh x + 2\sinh^3x##

##\sinh 3x= 2\sinh x + 2\sinh^3 x + \sinh x + 2\sinh^3x##

##\sinh 3x = 4\sinh^3x + 3 \sinh x##

Insight welcome...today i have been singing shine shine shine
Rather than repeatedly writing the left-hand side in every equation, I find it easier to read (and write) as a chain of equalities leading from the original left-hand side to the final right-hand expression.

IOW, like this:
##\sinh 3x ##
## =\sinh (2x+x)=\sinh 2x \cosh x + \cosh 2x \sinh x##
## =2\sinh x \cosh x \cosh x + (1+2 \sin^2x) \sinh x##
## = \dots ##
## = 4\sinh^3x + 3 \sinh x##

chwala and robphy
chwala said:
Homework Statement:: Prove the hyperbolic function corresponding to the given trigonometric function.

##\sin 3x = 3\sin x- 4\sin^3x##
Relevant Equations:: hyperbolic trig. properties.

Insight welcome...today i have been singing shine shine shine
Direct calculation from the formula
$$\sinh x=\frac{e^x-e^{-x}}{2}$$
$$\sin x=\frac{e^{ix}-e^{-ix}}{2i}$$
will give us the results including the different sign.

chwala said:
Homework Statement:: Prove the hyperbolic function corresponding to the given trigonometric function.

##\sin 3x = 3\sin x- 4\sin^3x##
Relevant Equations:: hyperbolic trig. properties.

We shall have;

##\sinh 3x = 3\sinh x- 4\sinh^3x## ...
In case it's not already clear, what you tried to prove:
##\sinh 3x = 3\sinh x- 4\sinh^3x##
is incorrect. It should be:
##\sinh 3x = 3\sinh x+ 4\sinh^3x##
which is what you correctly proved,

Normal and hyperbolic trig' identities don't necessarily match. For example ##cos^2x + sin^2x = 1## but ##cosh^2 – sinh^2x = 1##.

## 1. What is the equation "Shine Bright: Solving the Equation \sinh 3x = 4\sinh^3x + 3 \sinh x"?

The equation is a mathematical expression that involves the hyperbolic sine function (\sinh) and a variable (x). It states that the hyperbolic sine of 3x is equal to 4 times the hyperbolic sine cubed of x plus 3 times the hyperbolic sine of x.

## 2. What is the purpose of solving this equation?

The purpose of solving this equation is to find the value(s) of x that make the equation true. This can help in understanding the behavior of the hyperbolic sine function and can have applications in various fields such as physics, engineering, and economics.

## 3. Can this equation be solved analytically?

Yes, this equation can be solved analytically by using algebraic manipulation and trigonometric identities. However, the solution may involve complex numbers.

## 4. Are there any special techniques or strategies for solving this equation?

Yes, there are several techniques that can be used to solve this equation, such as substitution, factoring, and the quadratic formula. It may also be helpful to use a graphing calculator or computer software to visualize the equation and its solutions.

## 5. What are some possible applications of this equation in real life?

This equation can be used to model various physical phenomena, such as the motion of a damped harmonic oscillator or the shape of a hanging chain. It can also be applied in engineering for designing structures that can withstand certain forces or in economics for analyzing supply and demand curves.

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