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Chapter 16: Problem 16

Prove that \(\sinh 3 \theta=3 \sinh \theta+4 \sinh ^{3} \theta\)

Short Answer

Expert verified
To prove the identity \(\sinh 3 \theta=3 \sinh \theta+4 \sinh ^{3} \theta\), we first express \(\sinh 3 \theta\) as \(\sinh(2\theta+\theta)\) and use the addition formula for hyperbolic sine function. Then, we simplify the expression using properties of hyperbolic functions and factor out \(\sinh(\theta)\) from the expression. Finally, we compare the simplified expression to the given identity, and we can see that the simplification indeed shows the validity of the given identity: \(\sinh 3 \theta=3 \sinh\theta+4\sinh ^{3} \theta\).

Step by step solution

01

Definition of hyperbolic sin function

Recall the definition of the hyperbolic sine function, which is given by: \[\sinh(x) = \frac{e^{x} - e^{-x}}{2}\] We will use this definition to simplify the expression for \(\sinh 3 \theta\).
02

Use the addition formula for hyperbolic sine function

Recall the addition formula for hyperbolic sine function, which is: \[\sinh(A+B) = \sinh(A)\cosh(B) + \cosh(A)\sinh(B)\] Now, let's express \(\sinh 3 \theta\) as \(\sinh(2\theta+\theta)\). Then we can apply the addition formula with \(A=2\theta\) and \(B=\theta\): \[\sinh 3 \theta=\sinh(2\theta+\theta)=\sinh(2\theta)\cosh(\theta)+\cosh(2\theta)\sinh(\theta)\]
03

Simplify the expression using properties of hyperbolic functions

Recall the properties of hyperbolic functions: \[\sinh(2x)=2\sinh(x)\cosh(x)\] \[\cosh(2x)=\cosh^2(x)+\sinh^2(x)\] Using these identities, we can simplify the expressions for \(\sinh(2\theta)\) and \(\cosh(2\theta)\) in our equation: \[\sinh 3\theta= 2\sinh(\theta)\cosh(\theta)\cosh(\theta)+(\cosh^2(\theta)+\sinh^2(\theta))\sinh(\theta)\]
04

Further simplify the expression

Now let's simplify the expression further: \[\sinh 3\theta= 2\sinh(\theta)\cosh^2(\theta)+\sinh(\theta)(\cosh^2(\theta)+\sinh^2(\theta))\] Factor out \(\sinh(\theta)\) from the expression: \[\sinh 3\theta= \sinh(\theta)(2\cosh^2(\theta)+\cosh^2(\theta)+\sinh^2(\theta))\] Recall the property of hyperbolic functions: \[\cosh^2(x)-\sinh^2(x)=1\] Now, let's replace \(\cosh^2(\theta)-\sinh^2(\theta)=1\) by \(1 - \sinh^2(\theta) = \cosh^2(\theta)\) and simplify further: \[\sinh 3\theta= \sinh(\theta)(2(1-\sinh^2(\theta))+ 1 - \sinh^2(\theta) + \sinh^2(\theta))\] Simplify: \[\sinh 3\theta = \sinh(\theta)(3 + 4\sinh^2(\theta))\]
05

Compare the simplified expression to the given identity

Now, we can see that the simplified expression is indeed equal to the right-hand side of the given identity: \[\sinh 3\theta=3\sinh\theta+4\sinh^3\theta\] Thus, we have proved the given identity \(\sinh 3 \theta=3 \sinh\theta+4\sinh ^{3} \theta\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Trigonometric Identities
Understanding trigonometric identities is crucial when dealing with both standard trigonometric and hyperbolic functions. These identities are relationships between different functions that allow for simplification and transformation of expressions.

One commonly used identity in trigonometry is the Pythagorean identity, which for hyperbolic functions states that \(\cosh^2(x) - \sinh^2(x) = 1\). With this identity, one can transform expressions involving hyperbolic sine and cosine functions to make them more manageable. For example, expanding and simplifying expressions like \(\sinh(2x)\) becomes a much simpler task. In the exercise we considered, such identities were pivotal in proving the given hyperbolic sine function relationship.
Hyperbolic Functions Properties
Hyperbolic functions share many properties with their trigonometric counterparts, but they differ in certain aspects due to their definition through exponential functions. The hyperbolic sine function, \(\sinh(x)\), defined as \(\frac{e^{x} - e^{-x}}{2}\), exhibits properties that are essential to solving complex mathematical problems involving hyperbolae.

For instance, addition formulas for hyperbolic functions allow for the decomposition of \(\sinh(A+B)\) into \(\sinh(A)\cosh(B) + \cosh(A)\sinh(B)\), similar to the addition formulas in trigonometry. These properties are not just theoretical curiosities; they enable the simplification of hyperbolic expressions into forms that can be easily compared or integrated into more comprehensive mathematical models. As demonstrated in the exercise, by utilizing the properties of hyperbolic functions, we successfully transformed \(\sinh(3\theta)\) and proved its identity.
Proofs in Mathematics
Proofs are the cornerstone of mathematics, providing a way to establish the truth of a statement beyond any doubt. In this context, step-by-step logical reasoning is used to demonstrate why a particular mathematical statement must be true. Proofs can take many forms, such as direct proofs, proof by contradiction, or inductive proofs.

When approaching a proof, mathematicians often break down complex problems into simpler components, apply known identities or theorems, and carefully manipulate expressions to reveal new insights or relationships. The process involves a meticulous application of logic and mathematical knowledge, as we have seen with the proof of the hyperbolic sine function identity. By breaking down the expression and applying known hyperbolic function properties, the identity was shown to hold true, thus furnishing a clear example of a direct proof.

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Most popular questions from this chapter

Calculate from first principles, the value of: (a) \(\sinh ^{-1} 1.532\) (b) \(\cosh ^{-1} 1 \cdot 25\)

Prove that \(\cosh 2 x=1+2 \sinh ^{2} x\).

Express cosh \(2 x\) and \(\sinh 2 x\) in exponential form and hence solve for real values of \(x\), the equation: \(2 \cosh 2 x-\sinh 2 x=2\)

Prove that: (a) \(\sinh (x+y)=\sinh x \cosh y+\cosh x \sinh y\) (b) \(\cosh (x+y)=\cosh x \cosh y+\sinh x \sinh y\) Hence prove that: $$ \tanh (x+y)=\frac{\tanh x+\tanh y}{1+\tanh x \tanh y} $$

On the same axes, draw sketch graphs of (a) \(y=\sinh x\), (b) \(y=\cosh x\), (c) \(y=\tanh x\).
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