Problem 49 - College Trigonometry

Question

In Exercises 47 - 52 , we explore the hyperbolic cosine function, denoted \(\cosh (t)\), and the hyperbolic sine function, denoted \(\sinh (t)\), defined below: $$ \cosh (t)=\frac{e^{t}+e^{-t}}{2} \text { and } \sinh (t)=\frac{e^{t}-e^{-t}}{2} $$ Show that \(\\{x(t)=\cosh (t), y(t)=\sinh (t)\) parametrize the right half of the 'unit' hyperbola \(x^{2}-y^{2}=1 .\) (Hence the use of the adjective 'hyperbolic.')

Step-by-Step Solution

Answer

\(x(t) = \cosh(t)\) and \(y(t) = \sinh(t)\) parametrize the hyperbola \(x^2 - y^2 = 1\).

Step 1: Understanding the Given Functions

We start by recognizing the expressions for the hyperbolic functions: \(\cosh(t) = \frac{e^t + e^{-t}}{2}\) and \(\sinh(t) = \frac{e^t - e^{-t}}{2}\). Our task is to show that when \(x(t) = \cosh(t)\) and \(y(t) = \sinh(t)\), these parametrize the equation of the hyperbola \(x^2 - y^2 = 1\).

Step 2: Square the Hyperbolic Functions

Calculate \(x^2\) and \(y^2\) using the definitions of \(x(t)\) and \(y(t)\). We have:\[x^2 = \left(\frac{e^t + e^{-t}}{2}\right)^2 = \frac{(e^t + e^{-t})^2}{4}\]\[y^2 = \left(\frac{e^t - e^{-t}}{2}\right)^2 = \frac{(e^t - e^{-t})^2}{4}\]

Step 3: Expand and Simplify the Squares

Expand the squared terms:\[(e^t + e^{-t})^2 = e^{2t} + 2 + e^{-2t}\]\[(e^t - e^{-t})^2 = e^{2t} - 2 + e^{-2t}\]

Step 4: Subtract and Simplify

Subtract \(y^2\) from \(x^2\):\[x^2 - y^2 = \frac{e^{2t} + 2 + e^{-2t}}{4} - \frac{e^{2t} - 2 + e^{-2t}}{4}\]Simplify this expression:\[x^2 - y^2 = \frac{(e^{2t} + 2 + e^{-2t}) - (e^{2t} - 2 + e^{-2t})}{4} = \frac{4}{4} = 1\]

Step 5: Conclusion

Thus, \(x(t) = \cosh(t)\) and \(y(t) = \sinh(t)\) satisfy the equation \(x^2 - y^2 = 1\). This means these functions parametrically describe the right half of the unit hyperbola \(x^2 - y^2 = 1\).

Key Concepts

Unit Hyperbola

The unit hyperbola is a significant geometric concept in mathematics, particularly in hyperbolic functions. Unlike the unit circle in trigonometry, which is defined by the equation \(x^2 + y^2 = 1\), the unit hyperbola is defined using the equation \(x^2 - y^2 = 1\). This equation represents a hyperbola oriented along the x-axis in the Cartesian plane.
Visualize two opposing curves that are mirror images of each other across the y-axis. The part of the hyperbola in the first quadrant is of particular interest to us. This region, referred to as the "right half," can be represented using parametric functions.
The term "unit" in the unit hyperbola signifies that the hyperbola is centered at the origin and has vertices located at \((1, 0)\) and \((-1, 0)\). These parameters make it a fundamental object when illustrating hyperbolic functions and exploring the relationship between hyperbolic sine and cosine functions.

Parametric Equations

Parametric equations provide a convenient way to describe geometric figures using a parameter. In the context of the unit hyperbola, we use the hyperbolic functions \(x(t) = \cosh(t)\) and \(y(t) = \sinh(t)\) as parametric equations to describe the right half of the unit hyperbola.
Hyperbolic functions utilize exponential functions to establish their unique identities:\
\(x(t) = \frac{e^t + e^{-t}}{2}\) and \(y(t) = \frac{e^t - e^{-t}}{2}\)
This representation helps translate exponential growth and decay properties of the expressions into the spatial dimensions of the hyperbola. By plugging in different values of \(t\), you can find corresponding values for \(x\) and \(y\), effectively tracing out the shape of the hyperbola.
The beauty of parametric equations lies in their ability to encapsulate dynamic motion or change over time, useful in physics and animation, where describing motion is needed.

Hyperbolic Sine and Cosine

Hyperbolic sine and cosine, \(\sinh(t)\) and \(\cosh(t)\) respectively, are parallel to the sine and cosine functions from trigonometry. They express relationships between exponential functions and create foundational components for the unit hyperbola.
The definitions are as follows:

• \(\cosh(t) = \frac{e^t + e^{-t}}{2}\)
• \(\sinh(t) = \frac{e^t - e^{-t}}{2}\)

The hyperbolic cosine \(\cosh(t)\) represents the x-coordinate on the hyperbola by averaging exponential functions. Conversely, \(\sinh(t)\) signifies the y-coordinate through their difference.
These hyperbolic functions share a critical property used in forming the unit hyperbola: \(\cosh^2(t) - \sinh^2(t) = 1\), which mirrors the Pythagorean identity. It shows why \(x(t) = \cosh(t)\) and \(y(t) = \sinh(t)\) correctly parametrize the hyperbola, fulfilling the equation \(x^2 - y^2 = 1\). This link between exponential functions and hyperbolic geometry gives hyperbolic functions their distinctiveness and widespread utility in math and science.