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Applied trigonometry is fundamental in biomedical engineering, particularly when analyzing physiological signals, designing prosthetics, and interpreting imaging data. Understanding trigonometric functions allows engineers to model periodic phenomena such as heartbeats and respiratory cycles accurately. This chapter focuses on the essential trigonometric concepts and their practical application in biomedical engineering contexts in Kenya.
Trigonometry is critical in biomedical engineering for analyzing waveforms, designing biomedical devices, and solving spatial problems related to human anatomy. Kenyan biomedical engineers often use trigonometric principles when calibrating medical imaging equipment or developing biomechanical models. Mastery of trigonometric functions enables precise interpretation of angles and distances in three-dimensional biomedical systems.
Trigonometric functions model periodic biological signals such as ECG and EEG waves. The sine and cosine functions describe oscillations in electrical potentials generated by heart and brain activity.
Example 1: A heart signal waveform is modeled by \( y = 5 \sin(2 \pi t) \) where \( t \) is time in seconds. Find the amplitude at \( t = 0.25 \) s.
Given: Amplitude \(A = 5\), frequency \(f = 1\) Hz, \(t = 0.25\) s.
$$ y = 5 \sin(2 \pi \times 0.25) $$
$$ y = 5 \sin(\pi/2) $$
$$ y = 5 \times 1 $$
Answer: 5 \text{ units}
Example 2: The respiratory rate waveform is given by \( y = 3 \cos(4 \pi t) \). Calculate the value at \( t = 0.1 \) s.
Given: \(A = 3\), frequency \(f = 2\) Hz, \(t = 0.1\) s.
$$ y = 3 \cos(4 \pi \times 0.1) $$
$$ y = 3 \cos(0.4 \pi) $$
$$ y = 3 \times \cos(72^\circ) $$
$$ y = 3 \times 0.3090 $$
Answer: 0.927 \text{ units}
Example 3: Calculate the time when the ECG signal \( y = 6 \sin(5 \pi t) \) first reaches its maximum amplitude.
Given: \(A = 6\), angular frequency \(\omega = 5 \pi\).
Maximum occurs when \( \sin(\omega t) = 1 \), so
$$ \omega t = \frac{\pi}{2} $$
$$ 5 \pi t = \frac{\pi}{2} $$
$$ t = \frac{1}{10} = 0.1 \text{ s} $$
Answer: 0.1 \text{ seconds}
Trigonometry calculates joint angles and forces in prosthetic limb design and gait analysis. Kenyan engineers use these calculations to optimize device comfort and function.
Example 1: A prosthetic knee bends at an angle \( \theta \) such that \( \cos \theta = 0.6 \). Find the angle \( \theta \).
Given: \( \cos \theta = 0.6 \)
$$ \theta = \cos^{-1}(0.6) $$
$$ \theta = 53.13^\circ $$
Answer: 53.13^\circ
Example 2: The force \( F \) on a limb segment is resolved into horizontal and vertical components. If \( F = 200 \text{ N} \) and the angle with the horizontal is \( 30^\circ \), find the vertical component.
Given: \( F = 200 \text{ N} \), \( \theta = 30^\circ \)
$$ F_y = F \sin \theta $$
$$ F_y = 200 \times \sin 30^\circ $$
$$ F_y = 200 \times 0.5 $$
Answer: 100 \text{ N}
Example 3: Calculate the resultant force if two forces of 150 N and 100 N act at a \( 60^\circ \) angle to each other.
Given: \( F_1 = 150 \text{ N} \), \( F_2 = 100 \text{ N} \), \( \theta = 60^\circ \)
$$ R = \sqrt{F_1^2 + F_2^2 + 2 F_1 F_2 \cos \theta} $$
$$ R = \sqrt{150^2 + 100^2 + 2 \times 150 \times 100 \times 0.5} $$
$$ R = \sqrt{22500 + 10000 + 15000} $$
$$ R = \sqrt{47500} $$
$$ R = 217.94 \text{ N} $$
Answer: 217.94 \text{ N}
Trigonometry helps in reconstructing images from CT and MRI scans by calculating angles and distances in cross-sectional views. Kenyan biomedical engineers apply these principles in image processing algorithms.
Example 1: A CT scan slice forms a right triangle with sides 8 cm and 6 cm. Find the angle opposite the 6 cm side.
Given: Opposite side \( a = 6 \text{ cm} \), adjacent side \( b = 8 \text{ cm} \)
$$ \tan \theta = \frac{a}{b} $$
$$ \tan \theta = \frac{6}{8} = 0.75 $$
$$ \theta = \tan^{-1} (0.75) $$
$$ \theta = 36.87^\circ $$
Answer: 36.87^\circ
Example 2: The distance between two points in an MRI scan is found using the law of cosines. If sides are 5 cm and 7 cm with an included angle of \( 60^\circ \), find the third side.
Given: \( a = 5 \text{ cm} \), \( b = 7 \text{ cm} \), \( \theta = 60^\circ \)
$$ c^2 = a^2 + b^2 - 2ab \cos \theta $$
$$ c^2 = 5^2 + 7^2 - 2 \times 5 \times 7 \times 0.5 $$
$$ c^2 = 25 + 49 - 35 $$
$$ c^2 = 39 $$
$$ c = \sqrt{39} $$
$$ c = 6.24 \text{ cm} $$
Answer: 6.24 \text{ cm}
Example 3: Calculate the angle between two scan planes intersecting at 45° and 30° angles relative to a reference line.
Given: Angles \( \alpha = 45^\circ \), \( \beta = 30^\circ \)
$$ \theta = |\alpha - \beta| = |45^\circ - 30^\circ| = 15^\circ $$
Answer: 15^\circ
Trigonometric functions underpin Fourier analysis used in filtering biomedical signals to remove noise. Kenyan engineers apply this in ECG signal enhancement.
Example 1: A signal is represented by \( y = 4 \sin(3 \pi t) + 2 \cos(5 \pi t) \). Calculate its value at \( t = 0.2 \) s.
Given: \( y = 4 \sin(3 \pi t) + 2 \cos(5 \pi t) \), \( t = 0.2 \) s.
$$ y = 4 \sin(3 \pi \times 0.2) + 2 \cos(5 \pi \times 0.2) $$
$$ y = 4 \sin(0.6 \pi) + 2 \cos(\pi) $$
$$ y = 4 \times 0.9511 + 2 \times (-1) $$
$$ y = 3.804 - 2 $$
Answer: 1.804 \text{ units}
Example 2: Find the amplitude of the filtered signal \( y = 5 \cos(2 \pi t) \) at \( t = 0.25 \) s.
Given: \( y = 5 \cos(2 \pi t) \), \( t = 0.25 \) s.
$$ y = 5 \cos(2 \pi \times 0.25) $$
$$ y = 5 \cos(\pi/2) $$
$$ y = 5 \times 0 $$
Answer: 0 \text{ units}
Example 3: Determine the time when \( y = 3 \sin(4 \pi t) \) first reaches zero after \( t=0 \).
Given: \( y = 3 \sin(4 \pi t) \).
Zero crossing at \( \sin(4 \pi t) = 0 \), so
$$ 4 \pi t = n \pi, n=1,2,... $$
First zero after 0 when \( n=1 \):
$$ 4 \pi t = \pi $$
$$ t = \frac{1}{4} = 0.25 \text{ s} $$
Answer: 0.25 \text{ seconds}
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Create a free accountThis chapter explored the critical role of trigonometry in engineering, emphasizing its importance in solving practical problems. It began with the definitions of the six fundamental trigonometric ratios and functions, followed by an explanation of the unit circle and methods for measuring angles in degrees and radians. The chapter then examined the graphs of trigonometric functions, illustrating their periodic nature. A detailed study of trigonometric identities was presented, including fundamental, co-function, even-odd, sum and difference, double-angle, and half-angle formulas. Techniques for solving trigonometric equations were discussed, covering basic equations, the use of identities for simplification, and the application of inverse trigonometric functions. The chapter also highlighted engineering applications involving periodic functions, harmonic motion, and oscillations. Finally, trigonometry in triangles was addressed, focusing on solving right-angle and non-right-angle triangles using trigonometric ratios, the Law of Sines, and the Law of Cosines, with practical applications in structural analysis and navigation.
Calculate the sine, cosine, and tangent of a \(30^\circ\) angle used in positioning a biomedical sensor on an inclined plane. (2 marks)
A biomedical engineer measures an angle of \( \frac{\pi}{4} \) radians in a device joint. Find the corresponding angle in degrees. (2 marks)
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