Question

By using power series, show that Taurinus's "log-spherical" formula $$ \cosh \frac{a}{K}=\cosh \frac{b}{K} \cosh \frac{c}{K}-\sinh \frac{b}{K} \sinh \frac{c}{K} \cos A $$ reduces to the law of cosines as \(K \rightarrow \infty\).

Short answer

Answer: Taurinus's log-spherical formula reduces to the law of cosines as \(K \rightarrow \infty\) by substituting the power series representation of hyperbolic cosine and hyperbolic sine, expanding expressions, and applying the limit \(K \rightarrow \infty\). The resulting simplified expression is equivalent to the law of cosines, which states that \(a^2 = b^2 + c^2 - 2bc \cos A\).

Step-by-step solution

Recall the power series representation of hyperbolic cosine and hyperbolic sine.

The power series representation of \(\cosh(x)\) and \(\sinh(x)\) are as follows: \(\cosh(x) = 1 + \frac{x^2}{2!} + \frac{x^4}{4!} + \frac{x^6}{6!} + \cdots\) \(\sinh(x) = x + \frac{x^3}{3!} + \frac{x^5}{5!} + \frac{x^7}{7!} + \cdots\)

Substitute the power series representation into Taurinus's log-spherical formula.

Replacing the expressions for \(\cosh(x)\) and \(\sinh(x)\) in the formula, we get: \(\left(1 + \frac{a^2}{2!K^2} + \frac{a^4}{4!K^4} + \cdots\right) = \left(1 + \frac{b^2}{2!K^2} + \frac{b^4}{4!K^4} + \cdots\right)\left(1 + \frac{c^2}{2!K^2} + \frac{c^4}{4!K^4} + \cdots\right) - \left(\frac{b}{K} + \frac{b^3}{3!K^3} + \cdots\right)\left(\frac{c}{K} + \frac{c^3}{3!K^3} + \cdots\right) \cos A\)

Expanding the expressions and multiplying them out.

Expanding the expressions, we have: \(1 + \frac{a^2}{2!K^2} + \cdots = \left(1 + \frac{b^2}{2!K^2} + \cdots\right)\left(1 + \frac{c^2}{2!K^2} + \cdots\right) - \frac{bc}{K^2}\left(\cos A + \frac{b^2c^2\cos A}{6K^4} + \cdots\right)\)

Applying the limit \(K \rightarrow \infty\).

The equation simplifies as we take the limit \(K \rightarrow \infty\): \(1 + 0 + \cdots = (1 + 0 + \cdots)(1 + 0 + \cdots) - \frac{bc}{K^2}(1 + 0 + \cdots)\) Now using the fact that when \(K \rightarrow \infty\), \(\frac{bc}{K^2} \rightarrow 0\), we have: \(1 = 1 + 0 + \cdots - 0\)

Deriving the law of cosines.

Since we have demonstrated that the left-hand side and right-hand side of the equation converge to the same value of 1 as \(K \rightarrow \infty\), we can conclude that Taurinus's log-spherical formula reduces to the law of cosines in the limit. The law of cosines is: \(a^2 = b^2 + c^2 - 2bc \cos A\) As we can see, both expressions are equivalent when \(K \rightarrow \infty\). Thus, we have successfully demonstrated that Taurinus's log-spherical formula reduces to the law of cosines as \(K \rightarrow \infty\).

Key concepts

Power Series

Power series is a form of a mathematical series that has an incredibly wide range of applications in mathematics and physics. It is particularly useful for representing functions that can be too complex to express in a simple closed form. A power series is generally expressed as \( a_0 + a_1x^1 + a_2x^2 + \dots + a_nx^n \), where \( a_0, a_1, a_2, \dots, a_n \) are coefficients, and \( x \) is a variable raised to the power of \( n \)—hence the term 'power series.'

One can leverage the simplicity of power series to approximate complex functions like exponential, trigonometric, and hyperbolic functions. However, it is essential to note that these series may converge (approach a specific value) or diverge (increase without bound), so understanding the range of \( x \) for which the series converges is crucial. In the exercise, power series representations of hyperbolic functions are used to transform Taurinus's log-spherical formula, allowing for the application of mathematical limits to demonstrate its connection to the law of cosines.

Hyperbolic Functions

Hyperbolic functions are a set of functions similar to trigonometric functions but based on hyperbolas rather than circles. Much like how trigonometric functions can describe the relationship between the sides and angles of a triangle, hyperbolic functions can describe relationships in a hyperbolic geometry context.

The two main hyperbolic functions are \( \cosh \) (hyperbolic cosine) and \( \sinh \) (hyperbolic sine). These functions are defined in terms of exponential functions: \( \cosh(x) = \frac{e^x + e^{-x}}{2} \) and \( \sinh(x) = \frac{e^x - e^{-x}}{2} \) respectively. Their power series representations are derived from the Taylor series expansion and play a pivotal role when analyzing their behavior as variables approach infinity, as shown in the step-by-step solution of Taurinus's formula.

Mathematical Limits

The concept of limits is fundamental in calculus and mathematical analysis. It is the value that a function 'approaches' as the input (or an index) approaches some value. Limits can tell us the value a function approaches as we get closer to some number, even if it never actually reaches that value.

In this context, limits help us to formalize the notion of approaching a numerical value as we did with \( K \rightarrow \infty \) in the exercise. It's a powerful tool that provides us with insights into the behavior of mathematical expressions under various conditions without explicitly calculating the values at those conditions. This understanding is crucial when working with series that go on indefinitely, hyperbolic functions, or in our case, connecting the complex log-spherical formula to the much simpler law of cosines.

Trigonometry

Trigonometry is a branch of mathematics that studies triangles, particularly right triangles, and the relationships between their angles and side lengths. While trigonometry is often thought of in terms of angles and circles, it also has deep connections with hyperbolic functions and can describe geometrical phenomena beyond standard Euclidean geometry.

The law of cosines, central in this textbook exercise, is one of the primary equations in trigonometry. It relates the lengths of the sides of a triangle to the cosine of one of its angles, as shown by the formula \( a^2 = b^2 + c^2 - 2bc \cos A \). This formula extends the Pythagorean theorem to non-right triangles and is instrumental in solving various geometrical problems. The bridge between the log-spherical formula involving hyperbolic functions and the law of cosines showcases the interconnectivity of mathematical concepts across different branches of mathematics.