Engineering mathematics plays a critical role in Mechatronics Technology, providing the essential tools to analyze and design complex systems involving mechanical, electrical, and control components. Trigonometric and hyperbolic functions are fundamental in modelling periodic phenomena, waveforms, and system responses encountered in robotics, automation, and sensor technologies. This chapter focuses on applying these functions to solve practical problems relevant to Kenyan mechatronics engineers, enhancing their ability to interpret and manipulate angular relationships and hyperbolic behaviors in real-world applications.

2.1 Angles hyperbolic

Hyperbolic angles arise naturally in systems involving hyperbolic functions such as sinh, cosh, and tanh, which describe phenomena like cable shapes under gravity (catenaries), thermal distributions, and control system responses. Understanding hyperbolic angles and their properties is crucial for mechatronics professionals who model nonlinear system dynamics and signal processing tasks. This section covers the classification of hyperbolic angles into acute, obtuse, reflex, and right angles, with worked examples demonstrating their calculation and application.

2.1.1 Acute functions

Acute hyperbolic angles are those whose measure lies between 0 and \( \frac{\pi}{2} \) radians (0° and 90°). The hyperbolic sine, cosine, and tangent functions for acute angles are defined as:

$$ \sinh x = \frac{e^{x} - e^{-x}}{2}, \quad \cosh x = \frac{e^{x} + e^{-x}}{2}, \quad \tanh x = \frac{\sinh x}{\cosh x} $$

where \(x\) is the hyperbolic angle in radians.

Worked Examples

Example 1: Calculate the hyperbolic sine, cosine, and tangent of an acute hyperbolic angle \(x = 0.5\) radians.

Given: \(x = 0.5\) radians

$$\sinh 0.5 = \frac{e^{0.5} - e^{-0.5}}{2}$$

$$= \frac{1.6487 - 0.6065}{2}$$

$$= \frac{1.0422}{2} = 0.5211$$

$$\cosh 0.5 = \frac{e^{0.5} + e^{-0.5}}{2}$$

$$= \frac{1.6487 + 0.6065}{2}$$

$$= \frac{2.2552}{2} = 1.1276$$

$$\tanh 0.5 = \frac{\sinh 0.5}{\cosh 0.5} = \frac{0.5211}{1.1276} = 0.4621$$

Answer:

\(\sinh 0.5 = 0.5211\), \(\cosh 0.5 = 1.1276\), \(\tanh 0.5 = 0.4621\)

Example 2: A robotic arm joint rotates through a hyperbolic angle of 0.8 radians. Find the values of \(\sinh x\), \(\cosh x\), and \(\tanh x\).

Given: \(x = 0.8\) radians

$$\sinh 0.8 = \frac{e^{0.8} - e^{-0.8}}{2} = \frac{2.2255 - 0.4493}{2} = \frac{1.7762}{2} = 0.8881$$

$$\cosh 0.8 = \frac{e^{0.8} + e^{-0.8}}{2} = \frac{2.2255 + 0.4493}{2} = \frac{2.6748}{2} = 1.3374$$

$$\tanh 0.8 = \frac{0.8881}{1.3374} = 0.6640$$

Answer:

\(\sinh 0.8 = 0.8881\), \(\cosh 0.8 = 1.3374\), \(\tanh 0.8 = 0.6640\)

Example 3: A sensor signal varies according to the hyperbolic cosine of an acute angle \(x = 0.3\) radians. Calculate \(\cosh x\) and \(\tanh x\).

Given: \(x = 0.3\) radians

$$\cosh 0.3 = \frac{e^{0.3} + e^{-0.3}}{2} = \frac{1.3499 + 0.7408}{2} = \frac{2.0907}{2} = 1.0454$$

$$\sinh 0.3 = \frac{1.3499 - 0.7408}{2} = \frac{0.6091}{2} = 0.3045$$

$$\tanh 0.3 = \frac{0.3045}{1.0454} = 0.2913$$

Answer:

\(\cosh 0.3 = 1.0454\), \(\tanh 0.3 = 0.2913\)

2.1.2 Obtuse

Obtuse hyperbolic angles are those greater than \( \frac{\pi}{2} \) radians (90°) but less than \( \pi \) radians (180°). While hyperbolic functions are defined for all real numbers, interpreting obtuse hyperbolic angles often involves their relationship to the exponential functions governing hyperbolic sine and cosine.

The main formulas are the same as for acute angles:

$$ \sinh x = \frac{e^{x} - e^{-x}}{2}, \quad \cosh x = \frac{e^{x} + e^{-x}}{2}, \quad \tanh x = \frac{\sinh x}{\cosh x} $$

but \(x\) now lies in the range \( \frac{\pi}{2} < x < \pi \).

Worked Examples

Example 1: Calculate \(\sinh x\), \(\cosh x\), and \(\tanh x\) for \(x = 2\) radians (obtuse angle).

Given: \(x = 2\) radians

$$\sinh 2 = \frac{e^{2} - e^{-2}}{2} = \frac{7.3891 - 0.1353}{2} = \frac{7.2538}{2} = 3.6269$$

$$\cosh 2 = \frac{e^{2} + e^{-2}}{2} = \frac{7.3891 + 0.1353}{2} = \frac{7.5244}{2} = 3.7622$$

$$\tanh 2 = \frac{3.6269}{3.7622} = 0.9640$$

Answer:

\(\sinh 2 = 3.6269\), \(\cosh 2 = 3.7622\), \(\tanh 2 = 0.9640\)

Example 2: In a conveyor belt system, the hyperbolic angle related to the belt tension is \(x = 2.5\) radians. Find the hyperbolic sine and cosine.

Given: \(x = 2.5\) radians

$$\sinh 2.5 = \frac{e^{2.5} - e^{-2.5}}{2} = \frac{12.1825 - 0.0821}{2} = \frac{12.1004}{2} = 6.0502$$

$$\cosh 2.5 = \frac{e^{2.5} + e^{-2.5}}{2} = \frac{12.1825 + 0.0821}{2} = \frac{12.2646}{2} = 6.1323$$

Answer:

\(\sinh 2.5 = 6.0502\), \(\cosh 2.5 = 6.1323\)

Example 3: A robotic gripper angle corresponds to \(x = 3\) radians. Calculate \(\tanh x\).

Given: \(x = 3\) radians

$$\sinh 3 = \frac{e^{3} - e^{-3}}{2} = \frac{20.0855 - 0.0498}{2} = \frac{20.0357}{2} = 10.0179$$

$$\cosh 3 = \frac{e^{3} + e^{-3}}{2} = \frac{20.0855 + 0.0498}{2} = \frac{20.1353}{2} = 10.0677$$

$$\tanh 3 = \frac{10.0179}{10.0677} = 0.9951$$

Answer:

\(\tanh 3 = 0.9951\)

2.1.3 Reflex

Reflex hyperbolic angles exceed \( \pi \) radians (180°) but are less than \( 2\pi \) radians (360°). Though hyperbolic functions are defined for all real values, reflex angles in hyperbolic geometry often relate to periodic or cyclic phenomena in mechatronics such as repetitive signal waveforms or control cycles.

The formulas remain:

$$ \sinh x = \frac{e^{x} - e^{-x}}{2}, \quad \cosh x = \frac{e^{x} + e^{-x}}{2}, \quad \tanh x = \frac{\sinh x}{\cosh x} $$

for \( \pi < x < 2\pi \).

Worked Examples

Example 1: For a reflex hyperbolic angle \(x = 4\) radians, calculate \(\sinh x\), \(\cosh x\), and \(\tanh x\).

Given: \(x = 4\) radians

$$\sinh 4 = \frac{e^{4} - e^{-4}}{2} = \frac{54.5982 - 0.0183}{2} = \frac{54.5799}{2} = 27.2899$$

$$\cosh 4 = \frac{e^{4} + e^{-4}}{2} = \frac{54.5982 + 0.0183}{2} = \frac{54.6165}{2} = 27.3083$$

$$\tanh 4 = \frac{27.2899}{27.3083} = 0.9993$$

Answer:

\(\sinh 4 = 27.2899\), \(\cosh 4 = 27.3083\), \(\tanh 4 = 0.9993\)

Example 2: A sensor output varies according to \(\cosh x\) at \(x = 5\) radians. Calculate \(\cosh x\).

Given: \(x = 5\) radians

$$\cosh 5 = \frac{e^{5} + e^{-5}}{2} = \frac{148.4132 + 0.0067}{2} = \frac{148.4199}{2} = 74.2099$$

Answer:

\(\cosh 5 = 74.2099\)

Example 3: Calculate \(\tanh x\) for a hyperbolic angle \(x = 5.5\) radians.

Given: \(x = 5.5\) radians

$$\sinh 5.5 = \frac{e^{5.5} - e^{-5.5}}{2} = \frac{244.6919 - 0.0041}{2} = \frac{244.6878}{2} = 122.3439$$

$$\cosh 5.5 = \frac{e^{5.5} + e^{-5.5}}{2} = \frac{244.6919 + 0.0041}{2} = \frac{244.6960}{2} = 122.3480$$

$$\tanh 5.5 = \frac{122.3439}{122.3480} = 0.99996$$

Answer:

\(\tanh 5.5 = 0.99996\)

2.1.4 Right angle

The right hyperbolic angle corresponds to \(x = \frac{\pi}{2}\) radians (90°). Evaluating hyperbolic sine, cosine, and tangent at this angle is important in mechatronics when analyzing orthogonal components in system responses or signal processing.

The functions are:

$$ \sinh \frac{\pi}{2} = \frac{e^{\frac{\pi}{2}} - e^{-\frac{\pi}{2}}}{2}, \quad \cosh \frac{\pi}{2} = \frac{e^{\frac{\pi}{2}} + e^{-\frac{\pi}{2}}}{2}, \quad \tanh \frac{\pi}{2} = \frac{\sinh \frac{\pi}{2}}{\cosh \frac{\pi}{2}} $$

Worked Examples

Example 1: Calculate \(\sinh \frac{\pi}{2}\), \(\cosh \frac{\pi}{2}\), and \(\tanh \frac{\pi}{2}\).

Given: \(x = \frac{\pi}{2} = 1.5708\) radians

$$\sinh 1.5708 = \frac{e^{1.5708} - e^{-1.5708}}{2} = \frac{4.8105 - 0.2079}{2} = \frac{4.6026}{2} = 2.3013$$

$$\cosh 1.5708 = \frac{e^{1.5708} + e^{-1.5708}}{2} = \frac{4.8105 + 0.2079}{2} = \frac{5.0184}{2} = 2.5092$$

$$\tanh 1.5708 = \frac{2.3013}{2.5092} = 0.9160$$

Answer:

\(\sinh \frac{\pi}{2} = 2.3013\), \(\cosh \frac{\pi}{2} = 2.5092\), \(\tanh \frac{\pi}{2} = 0.9160\)

Example 2: A control signal is proportional to \(\tanh \frac{\pi}{2}\). Calculate its value.

Given: \(x = \frac{\pi}{2} = 1.5708\) radians

Using previous result:

$$\tanh 1.5708 = 0.9160$$

Answer:

\(\tanh \frac{\pi}{2} = 0.9160\)

Example 3: Determine \(\cosh \frac{\pi}{2}\) for use in a mechatronic system involving hyperbolic functions.

Given: \(x = \frac{\pi}{2} = 1.5708\) radians

From above:

$$\cosh 1.5708 = 2.5092$$

Answer:

\(\cosh \frac{\pi}{2} = 2.5092\)

Chapter Summary

This chapter covered the application of trigonometric and hyperbolic functions starting with the classification of angles including acute, obtuse, reflex, and right angles. It then examined different types of triangles such as isosceles, equilateral, right angled, and scalene, highlighting their unique properties. The fundamental trigonometric ratios sine, cosine, tangent, cosecant, secant, and cotangent were explained and applied to solve problems. The chapter also introduced key trigonometric identities, including proofs and the essential Pythagorean identities, which are critical for simplifying expressions. Trigonometric equations were explored with methods for finding their solutions. Finally, the chapter presented hyperbolic functions including sinh, cosh, cosech, tanh, and sech, detailing their definitions and uses in engineering contexts. Throughout, emphasis was placed on practical application of these functions in solving engineering mathematics problems.