By the end of this chapter, you will be able to:
Mastering these skills will help you tackle real-world engineering challenges with precision and confidence.
Engineering mathematics is fundamental to biomedical engineering, providing the tools to model, analyze, and solve problems involving biological systems and medical devices. Trigonometry, in particular, enables the analysis of angles, distances, and periodic phenomena encountered in biomedical instrumentation, imaging, and biomechanics. This chapter develops essential trigonometric skills applied in biomedical engineering contexts such as joint angle measurement, ECG signal interpretation, and prosthetic design.
Trigonometry is crucial in biomedical engineering to quantify spatial relationships and waveforms that arise in human physiology and medical technology. Accurate angle measurements support gait analysis and orthopaedic assessments, while trigonometric functions model periodic biological signals such as heartbeats and neural oscillations. Kenyan biomedical engineers use trigonometry to design devices like ultrasound probes and robotic surgical arms, where precise angular positioning is vital.
Biomedical signals such as ECG, EEG, and EMG exhibit periodic waveforms that can be described using sine and cosine functions. Trigonometry helps analyze these signals by decomposing them into fundamental frequency components, enabling noise filtering and feature extraction.
Example 1: An ECG signal has a frequency of 1 Hz. Calculate the period of the signal.
Given: \( f = 1 \, \text{Hz} \)
Formula: \( T = \frac{1}{f} \)
Substitution: \( T = \frac{1}{1} \)
Calculation: \( T = 1 \, \text{second} \)
Answer: 1 second
Example 2: The amplitude of a sinusoidal EMG signal is 5 mV. Find the value of the signal at \(30^\circ\).
Given: \( A = 5 \, \text{mV}, \theta = 30^\circ \)
Formula: \( V = A \sin \theta \)
Substitution: \( V = 5 \sin 30^\circ \)
Calculation: \( V = 5 \times 0.5 = 2.5 \, \text{mV} \)
Answer: 2.5 mV
Example 3: A neural oscillation is modeled as \( V = 3 \cos \theta \). Calculate the voltage at \( 120^\circ \).
Given: \( A = 3, \theta = 120^\circ \)
Formula: \( V = A \cos \theta \)
Substitution: \( V = 3 \cos 120^\circ \)
Calculation: \( V = 3 \times (-0.5) = -1.5 \)
Answer: -1.5 units
Kenyan Context Example:
At Kenyatta National Hospital, biomedical engineers analyze ECG signals using trigonometric functions to filter noise and extract heart rate information. For instance, when calibrating ECG monitors supplied by Mindray Kenya Ltd, engineers use the period and frequency calculations above to ensure accurate readings.
Trigonometry allows calculation of joint angles and limb positions, essential for designing prosthetics and orthotic devices that replicate natural movement. Measuring angles between bones or components ensures proper fitting and function.
Example 1: Calculate the angle between the thigh and shin if the vertical distance is 40 cm and horizontal distance is 30 cm.
Given: Opposite = 30 cm, Adjacent = 40 cm
Formula: \( \theta = \tan^{-1} ( \frac{\text{Opposite}}{\text{Adjacent}} ) \)
Substitution: \( \theta = \tan^{-1} ( \frac{30}{40} ) \)
Calculation: \( \theta = \tan^{-1} (0.75) = 36.87^\circ \)
Answer: 36.87 degrees
Example 2: A prosthetic limb segment is 50 cm long. Calculate the vertical height if it is inclined at \( 25^\circ \).
Given: Length = 50 cm, Angle = \( 25^\circ \)
Formula: \( \text{Height} = L \sin \theta \)
Substitution: \( \text{Height} = 50 \sin 25^\circ \)
Calculation: \( \text{Height} = 50 \times 0.4226 = 21.13 \, \text{cm} \)
Answer: 21.13 cm
Example 3: Determine the horizontal distance covered by a robotic arm segment 60 cm long inclined at \( 40^\circ \).
Given: Length = 60 cm, Angle = \( 40^\circ \)
Formula: \( \text{Horizontal distance} = L \cos \theta \)
Substitution: \( \text{Horizontal distance} = 60 \cos 40^\circ \)
Calculation: \( \text{Horizontal distance} = 60 \times 0.7660 = 45.96 \, \text{cm} \)
Answer: 45.96 cm
Kenyan Context Example:
At the Kenya Medical Research Institute (KEMRI), prosthetic devices are designed using trigonometric calculations to ensure proper limb alignment and function. Engineers measure joint angles and limb segments for patients using locally manufactured prosthetics by Nairobi Prosthetics Ltd.
Trigonometric functions help calibrate imaging devices like ultrasound and CT scanners by determining probe angles and correcting distortions. Accurate angle measurements improve image resolution and diagnostic accuracy.
Example 1: An ultrasound probe is tilted at \( 15^\circ \) from the vertical. Calculate the horizontal displacement of the beam after penetrating 10 cm.
Given: Distance = 10 cm, Angle = \( 15^\circ \)
Formula: \( d = L \sin \theta \)
Substitution: \( d = 10 \sin 15^\circ \)
Calculation: \( d = 10 \times 0.2588 = 2.588 \, \text{cm} \)
Answer: 2.59 cm
Example 2: A CT scanner rotates through \( 270^\circ \). Convert this angle to radians.
Given: \( \theta = 270^\circ \)
Formula: \( \theta_{\text{radians}} = \theta^\circ \times \frac{\pi}{180} \)
Substitution: \( \theta_{\text{radians}} = 270 \times \frac{\pi}{180} \)
Calculation: \( \theta_{\text{radians}} = 1.5 \pi \, \text{radians} \)
Answer: \(1.5 \pi\) radians
Example 3: Calculate the arc length of the probe path if the radius is 8 cm and the angle is \( 60^\circ \).
Given: Radius = 8 cm, Angle = \( 60^\circ \)
Formula: \( s = r \theta_{\text{radians}} \)
Substitution:
Convert angle: \( 60^\circ \times \frac{\pi}{180} = \frac{\pi}{3} \)
\( s = 8 \times \frac{\pi}{3} \)
Calculation: \( s = \frac{8 \pi}{3} = 8.38 \, \text{cm} \)
Answer: 8.38 cm
Kenyan Context Example:
Biomedical engineers at Aga Khan University Hospital calibrate ultrasound and CT scanners using trigonometric calculations to optimize probe angles and improve image quality. Devices from GE Healthcare Kenya are routinely checked for angular accuracy using the methods described above.
Create a free account to open more of this chapter.
Free: practical guides, quick cards, workplace scenarios and more.
Create a free accountThis chapter explored the fundamental role of trigonometry in engineering, highlighting its importance in solving practical problems. It began with the definitions of the six primary trigonometric functions and introduced the unit circle as a basis for understanding angle measurements in degrees and radians, along with the graphical representation of these functions. The chapter then covered essential trigonometric identities, including fundamental, Pythagorean, reciprocal, quotient, co-function, and even-odd properties, as well as advanced formulas such as sum and difference, double-angle, and half-angle identities. Methods for solving trigonometric equations were examined, emphasizing the use of basic equations, identities, and inverse functions, with applications to periodic phenomena and harmonic motion in engineering systems. The discussion extended to solving triangles, focusing on right-angle triangles through trigonometric ratios and engineering applications like inclined planes and forces, then addressing non-right-angle triangles using the Law of Sines and Law of Cosines. Finally, the chapter demonstrated how trigonometry is applied in fields such as structural analysis and navigation, underscoring its practical value across engineering disciplines.
At the start of this chapter we promised you would be able to:
Tick each one you can genuinely do.
So, are you there yet?
You're competent when you can confidently do 50% or more of what this chapter promised.
Sign in to record how you're doing.