By the end of this chapter, you will be able to:
Mastering these skills will help you solve real-world engineering problems with precision and confidence, making you a valuable technician in your field.
Engineering technicians in electronics frequently encounter problems involving angles, waveforms, and signal analysis where trigonometric and hyperbolic functions play a crucial role. Understanding these functions enables accurate design, troubleshooting, and optimization of circuits and systems such as oscillators, filters, and communication devices. This chapter develops your ability to apply trigonometry and hyperbolic functions in typical electronics engineering contexts encountered in Kenya.
Trigonometry is fundamental in electronics engineering as it allows calculation and analysis of alternating currents, voltages, and signal phases. Engineers rely on trigonometric functions to describe sinusoidal waveforms, analyze phase shifts in AC circuits, and design components like antennas and filters. Mastery of trigonometry enables technicians to model and solve problems involving waves, oscillations, and vector quantities encountered in devices used across Kenyan industries such as telecommunications, power distribution, and manufacturing.
Alternating current (AC) circuits produce sinusoidal voltages and currents described by trigonometric functions. The voltage \(V\) and current \(I\) in an AC circuit vary as functions of time \(t\), often expressed as:
$$ V = V_m \sin(\omega t + \phi) $$
where \(V_m\) is the maximum voltage, \(\omega\) is angular frequency, and \(\phi\) is phase angle.
Example 1: Calculate the instantaneous voltage at time \(t = 0.01\) s for an AC voltage with \(V_m = 230\) V, frequency \(f = 50\) Hz, and zero phase angle.
Given: \(V_m = 230\) V, \(f = 50\) Hz, \(t = 0.01\) s, \(\phi = 0\)
$$ \omega = 2 \pi f = 2 \times 3.142 \times 50 = 314.2 \text{ rad/s} $$
$$ V = 230 \sin(314.2 \times 0.01 + 0) = 230 \sin(3.142) $$
$$ V = 230 \times 0 = 0 \text{ V} $$
Answer: 0 V
Example 2: Find the voltage at \(t = 0.005\) s with the same values.
$$ V = 230 \sin(314.2 \times 0.005) = 230 \sin(1.571) $$
$$ V = 230 \times 1 = 230 \text{ V} $$
Answer: 230 V
Example 3: Calculate the instantaneous voltage at \(t = 0.007\) s if the phase angle \(\phi = 30^\circ\) (or \(\pi/6\) radians).
$$ V = 230 \sin(314.2 \times 0.007 + \frac{\pi}{6}) = 230 \sin(2.199 + 0.524) $$
$$ V = 230 \sin(2.723) = 230 \times 0.403 = 92.69 \text{ V} $$
Answer: 92.69 V
Trigonometry describes waveforms in signal processing, enabling calculation of amplitude, frequency, and phase shifts. Engineers analyze sine and cosine components to understand modulation and filtering effects in communication systems.
Example 1: Determine the amplitude of a signal \(S(t) = 5 \cos(2 \pi \times 1000 t)\).
Given: \(S(t) = 5 \cos(2 \pi \times 1000 t)\)
Amplitude is the coefficient of cosine.
Answer: 5 units
Example 2: A signal is given by \(S(t) = 10 \sin(2000 \pi t + \frac{\pi}{4})\). Find the phase angle in degrees.
Given: \(\phi = \frac{\pi}{4}\) radians
$$ \phi = \frac{\pi}{4} \times \frac{180}{\pi} = 45^\circ $$
Answer: 45 degrees
Example 3: Calculate the value of the signal at \(t = 0.0005\) s.
$$ S = 10 \sin(2000 \pi \times 0.0005 + \frac{\pi}{4}) = 10 \sin(3.142 + 0.785) $$
$$ S = 10 \sin(3.927) = 10 \times (-0.707) = -7.07 $$
Answer: -7.07 units
Calculations of angles of incidence, reflection, and radiation patterns use trigonometry to optimize antenna performance. This is critical in Kenya’s expanding telecommunications and broadcasting sectors.
Example 1: An antenna radiates at an angle of \(60^\circ\). Calculate the horizontal component of a 100 m signal length.
Given: Length \(L = 100\) m, angle \(\theta = 60^\circ\)
$$ L_{horizontal} = L \cos \theta = 100 \times \cos 60^\circ $$
$$ L_{horizontal} = 100 \times 0.5 = 50 \text{ m} $$
Answer: 50 m
Example 2: Find the vertical component of the same signal length.
$$ L_{vertical} = L \sin \theta = 100 \times \sin 60^\circ $$
$$ L_{vertical} = 100 \times 0.866 = 86.6 \text{ m} $$
Answer: 86.6 m
Example 3: If the antenna rotates by \(30^\circ\), calculate the new horizontal component.
$$ L_{horizontal} = 100 \times \cos 30^\circ = 100 \times 0.866 = 86.6 \text{ m} $$
Answer: 86.6 m
Calculate the instantaneous voltage of an AC supply with \(V_m = 120\) V, frequency 60 Hz, and phase angle \(45^\circ\) at \(t = 0.002\) s. (5 marks)
A signal is represented by \(S(t) = 8 \cos(2 \pi \times 500 t + \frac{\pi}{3})\). Find the amplitude and phase angle in degrees. (6 marks)
An antenna has a length of 150 m and radiates at \(45^\circ\). Determine the vertical and horizontal components of the signal. (6 marks)
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Create a free accountThis chapter explored the critical role of trigonometry in engineering, beginning with the fundamental trigonometric ratios and functions such as sine, cosine, tangent, and their reciprocals. It examined the unit circle and the measurement of angles in degrees and radians, alongside the graphical representation of these functions. The chapter detailed essential trigonometric identities, including Pythagorean, reciprocal, quotient, co-function, and sum and difference formulas, which are vital for simplifying expressions and solving equations. Methods for solving basic and complex trigonometric equations were presented, incorporating inverse functions and their application to engineering problems involving periodic phenomena and harmonic motion. The use of trigonometry in solving triangles, both right-angled and oblique, was discussed with practical applications in structural analysis and navigation. An introduction to hyperbolic functions covered their definitions, graphs, and key identities analogous to trigonometric ones. Finally, techniques for solving hyperbolic equations and the use of inverse hyperbolic functions were explained, concluding with their engineering applications in various technical contexts.
Calculate the sine, cosine, and tangent of a \(30^\circ\) angle in an electronics circuit where a signal is incident at this angle on a sensor. (2 marks)
A voltage waveform is described by the function \(v(t) = 5 \sin(100 \pi t)\). Calculate the voltage at \(t = 0.01\) seconds. (2 marks)
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