Question

Show that Taurinus's formula for an asymptotic right triangle on a sphere of imaginary radius \(i\), namely, \(\sin B=\) \(1 / \cosh x\), is equivalent to Lobachevsky's formula for the angle of parallelism, \(\tan \frac{B}{2}=e^{-x}\).

Short answer

In this exercise, we showed that Taurinus's formula, which describes the sine of an angle in a right triangle on a sphere with an imaginary radius i, is equivalent to Lobachevsky's formula for the angle of parallelism. We used trigonometric identities and properties of hyperbolic functions to express sine in terms of hyperbolic functions, then we used the double-angle formula to find the tangent of half the angle, and finally, after substituting the expressions for sine and cosine, we were able to demonstrate the equivalence of both formulas.

Step-by-step solution

Write down Taurinus's formula

Taurinus's formula for an asymptotic right triangle on a sphere of imaginary radius i is given as: $$ \sin B = \frac{1}{\cosh x} $$ We need to show that this is equivalent to Lobachevsky's formula for the angle of parallelism: $$ \tan \frac{B}{2} = e^{-x} $$

Express the sine in terms of hyperbolic functions

To work with the hyperbolic functions, we will use the following trigonometric identity to express sine in terms of the hyperbolic sine (sinh) and cosine (cosh): $$ \sin B = \frac{\sinh(x)}{\cosh^2(x)} $$

Find the tangent of half the angle using the double-angle formula

Now we want to find \(\tan \frac{B}{2}\), and to do this, we will use the double-angle formula for tangent, which is: $$ \tan \frac{B}{2} = \frac{\sin B}{1 + \cos B} $$ We will then substitute our expression for \(\sin B\) from Step 2 into the double-angle formula.

Substitute the expression for sine

Plugging the expression for \(\sin B\) from Step 2 into the double-angle formula, we get: $$ \tan \frac{B}{2} = \frac{\frac{\sinh(x)}{\cosh^2(x)}}{1 + \cos B} $$

Express the cosine in terms of hyperbolic functions and substitute

We know that \(\cos^2 B = 1 - \sin^2 B\). Using the given Taurinus's formula, we have \(\sin^2 B = \frac{1}{\cosh^2 x}\). Thus, we can write \(\cos^2 B = 1 - \frac{1}{\cosh^2 x}\) and find \(\cos B = \frac{\sinh x}{\cosh x}\). Now, we can substitute this into the formula for \(\tan \frac{B}{2}\).

Simplify the expression and show equivalence

After substituting the expressions for \(\sin B\) and \(\cos B\), we find: $$ \tan \frac{B}{2} = \frac{\frac{\sinh(x)}{\cosh^2(x)}}{1 + \frac{\sinh x}{\cosh x}} $$ Multiplying both numerator and denominator by \(\cosh x\), we get: $$ \tan \frac{B}{2} = \frac{\sinh(x)}{\cosh x + \sinh x} $$ Now we observe that the denominator can be written as a hyperbolic function, namely \(\cosh(x)\): $$ \cosh x + \sinh x = e^x $$ Substituting this back into our expression for \(\tan \frac{B}{2}\), we obtain: $$ \tan \frac{B}{2} = \frac{\sinh(x)}{e^x} $$ Finally, using the property \(\sinh(x) = \frac{e^x - e^{-x}}{2}\), we get: $$ \tan \frac{B}{2} = \frac{\frac{e^x - e^{-x}}{2}}{e^x} = \frac{1 - e^{-2x}}{2e^x} $$ Now, expressing the numerator as the sum of two fractions: $$ \tan \frac{B}{2} = \frac{e^{2x} - 1}{2e^{2x}} $$ And finally, dividing both numerator and denominator by \(e^{2x}\), we obtain the desired result: $$ \tan \frac{B}{2} = e^{-x} $$ This demonstrates the equivalence of Taurinus's formula and Lobachevsky's formula for the angle of parallelism.

Key concepts

Hyperbolic Functions

Hyperbolic functions are the analogs of the ordinary trigonometric functions defined for a hyperbola rather than on a circle. Just as trigonometric functions can describe the relationships between the sides and angles of a right triangle, hyperbolic functions are useful in the study of theoretical concepts like time dilation in relativity and the geometry of hyperbolic spaces.

There are several basic hyperbolic functions, including the hyperbolic sine \(\sinh x\), hyperbolic cosine \(\cosh x\), and hyperbolic tangent \(\tanh x\). These functions are defined using exponential functions. For instance, \(\sinh x = \frac{e^x - e^{-x}}{2}\) and \(\cosh x = \frac{e^x + e^{-x}}{2}\), which reveals their relation to the exponential function and how they can be expressed through it.

Hyperbolic functions are not just mathematical curiosities; they appear in various areas of mathematics and physics, such as the shape of a hanging cable, known as a catenary, or in the solution to the Laplace equation in Cartesian coordinates.

Angle of Parallelism

The angle of parallelism is a concept from hyperbolic geometry, a non-Euclidean geometry that denies the parallel postulate stating that for any given line and a point not on the line, there is only one line through the point that does not intersect the original line. In hyperbolic geometry, there are infinitely many such lines, and the angle of parallelism describes how these lines diverge.

The angle of parallelism \(\Pi(x)\) for a given distance 'x' from a point to a line is the angle at which a line segment originating at the point must be inclined to ensure that it remains parallel – that is, it never meets – the given line. Taurinus's formula relates to this concept by specifying the relationship of the angle of parallelism in terms of hyperbolic functions of the distance 'x'.

This concept highlights the unique nature of hyperbolic geometry and has implications in understanding the geometry of the universe and areas like relativistic physics, where Euclidean geometry doesn't always apply.

Lobachevsky's Formula

Lobachevsky's formula is a direct application of the angle of parallelism in hyperbolic geometry, named after the Russian mathematician Nikolai Lobachevsky. It provides a method to compute the angle of parallelism given a distance 'x' in a hyperbolic plane. This formula is noteworthy because it encapsulates the essence of how hyperbolic geometry diverges from Euclidean perspectives.

The formula states that the tangent of half the angle of parallelism equals the negative natural exponentiation of the distance, expressed as \(\tan \frac{B}{2} = e^{-x}\). This concise relationship allows for the computation of angles in a hyperbolic space and shows the exponential nature of divergence of 'parallel' lines in this non-Euclidean framework.

Understanding Lobachevsky's formula and its implications requires a deep dive into the fundamentals of hyperbolic functions, as demonstrated in the step-by-step solution of Taurinus's formula. It underscores the profound connections between geometry, algebra, and analysis in mathematical theory.