Result

Calculation Steps

Visual Representation

About Trigonometric and Hyperbolic Functions

What are Trigonometric and Hyperbolic Functions?

Trigonometric functions relate angles to the sides of a right triangle, while hyperbolic functions are analogous to trigonometric functions but based on hyperbolas instead of circles.

Formulas and Their Meanings

1. Cosine function: \(f(x) = \cos(x)\)

• Represents the x-coordinate on a unit circle for a given angle x
• Period: 2π
• Range: [-1, 1]

2. Secant function: \(f(x) = \sec(x) = \frac{1}{\cos(x)}\)

• Reciprocal of cosine
• Period: 2π
• Range: (-∞, -1] ∪ [1, ∞)

3. Hyperbolic cosine function: \(f(x) = \cosh(x) = \frac{e^x + e^{-x}}{2}\)

• Related to the regular cosine but based on exponential functions
• No period (not periodic)
• Range: [1, ∞)

Calculation Steps

1. Define the functions: \(f_1(x) = \cos(x)\), \(f_2(x) = \sec(x)\), and \(f_3(x) = \cosh(0.25x)\)
2. Choose a range of x-values
3. Evaluate all three functions for each x-value
4. Plot the results on the same coordinate system

Example

Let's evaluate all three functions at x = π/4:

1. \(f_1(\frac{\pi}{4}) = \cos(\frac{\pi}{4}) = \frac{\sqrt{2}}{2} \approx 0.7071\)

2. \(f_2(\frac{\pi}{4}) = \sec(\frac{\pi}{4}) = \frac{1}{\cos(\frac{\pi}{4})} = \sqrt{2} \approx 1.4142\)

3. \(f_3(\frac{\pi}{4}) = \cosh(0.25 \cdot \frac{\pi}{4}) = \cosh(\frac{\pi}{16}) \approx 1.0491\)