Chapter 4: Apply trigonometric functions

Many signals and movements that a biomedical engineering technician meets repeat or turn through an angle: the waveform on a patient monitor, the swing of a joint in a rehabilitation device, the tilt of an adjustable support frame. Trigonometric functions describe these angles and waves by linking the sides and angles of a right-angled triangle to the repeating sine and cosine curves. This chapter introduces the trigonometric ratios and functions and their graphs, the identities, the solving of trigonometric equations and the use of triangles, including the sine and cosine rules, to find unknown sides and angles.

4.1 Importance of trigonometry in engineering.

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.

Role in signal analysis

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.

Worked Examples

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: \(0.1 \text{ seconds}\)

Application in biomechanics

Trigonometry calculates joint angles and forces in prosthetic limb design and gait analysis. Kenyan engineers use these calculations to optimize device comfort and function.

Worked Examples

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: \(217.94 \text{ N}\)

Use in medical imaging

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.

Worked Examples

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: \(15^\circ\)

Signal processing and filtering

Trigonometric functions underpin Fourier analysis used in filtering biomedical signals to remove noise. Kenyan engineers apply this in ECG signal enhancement.

Worked Examples

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: \(0.25 \text{ seconds}\)

Chapter Summary

This chapter showed how a biomedical engineering technician uses trigonometry to describe angles, waves and triangles. It began with the uses of trigonometry in the trade: modelling repeating signals, finding joint angles and resolving forces. The six trigonometric ratios were defined from a right-angled triangle, extended to any angle with the unit circle, and the measure of an angle was changed between degrees and radians. The graphs of the sine, cosine and tangent functions were described by their amplitude, period and phase shift. The identities followed: the Pythagorean, reciprocal and quotient identities, the co-function and even-odd properties, and the sum, difference, double-angle and half-angle formulas. Trigonometric equations were then solved with reference angles, with identities that reduce them to one ratio, and with the inverse functions, and the same methods were applied to waveforms and to the harmonic motion of a mass on a spring. Finally, right-angled triangles were solved with the three ratios, including inclined planes and the components of a force, and oblique triangles were solved with the law of sines and the law of cosines, including the ambiguous case, structural frames and bearings.