The basic trigonometric functions sine and cosine are often used to model light, sound, and electromagnetic waves. They can also be used to approximate 1 other periodic phenomena, such as tides or blood pressure.
Understanding these functions (and their individual parts) allows us to effectively predict and analyze periodic phenomena and their properties. For example, we can calculate the height of ocean waves or the time of high tide. In the following text, we will focus on functions approximating blood pressure.
The heart acts as a pump that pushes oxygenated blood through the blood vessels throughout the body to provide it with the oxygen and nutrients it needs. Blood pressure is the pressure that blood exerts on the walls of the blood vessels it flows through. This pressure varies in different parts of the bloodstream. Blood pressure is commonly referred to as arterial blood pressure, the pressure of blood in the large arteries. If your blood pressure is too high, it puts strain on your arteries (and heart), which can lead to a heart attack or stroke.
High blood pressure, also known as hypertension, is something you usually don’t feel or notice. It doesn’t tend to cause any obvious signs or symptoms. The only way to know what your blood pressure is is to have it measured. The first blood pressure monitor measured blood pressure based on the height of the mercury column, which is why it is still measured in millimeters of mercury. It is written as a ratio of two numbers. For example, if your reading is \(120/80\,\text{mm}\,Hg\), your blood pressure is called \(120\) over \(80\).
A higher number indicates systolic blood pressure. This is the highest value that your blood pressure reaches when your heart is beating and pushing blood into the vascular system. The lower number is called diastolic blood pressure and is, in contrast, the lowest level, the pressure reaches when the heart muscle relaxes between beats.
The blood pressure chart below shows the ranges of high, low, and healthy blood pressure.
During heart contractions, blood pressure increases and decreases cyclically. The duration of one heartbeat corresponds to the period of the function that represents blood pressure. Each period of the blood pressure measurement function corresponds to one heartbeat (indicating how long it takes for a heartbeat cycle to complete). In addition, we know that the local maximum of the function will be the systolic blood pressure value and the local minimum will be the diastolic pressure value. We can approximately replace the blood pressure function with the sine or cosine function. Recall that the general prescription of the sine function is
\[ f\left(x\right) = a\cdot\sin\left(bx + c\right) +d, \]
where \(a\) represents the amplitude, which determines how high or low the sinusoid moves from its mean position. The value of \(b\) affects the period of the function (the period is equal to \(\frac{2\pi}{|b|}\).). The term \(c\) determines the shift in the direction of the \(x\) axis and \(d\) the shift in the direction of the \(y\) axis.
Exercise 1. Compare the following two functions \[P_1(t)=25\cdot\sin\left(\frac{7\pi}{3}t\right)+105,\quad P_2(t)=30\cdot\cos\left(2\pi t\right)+125,\] which approximate the blood pressure of two different people. These functions depend on the variable \(t\), which represents time in seconds. For each function, find the period of the function (the length of one heartbeat) and determine their heart rate (the number of heartbeats per minute).
Solution. The period \(p_1\) of the function \(P_1\) can be calculated as \[p_1=\frac{2\pi}{\frac{7\pi}{3}}=\frac{6}{7}\,\text{sekundy}.\] Since the duration of one heartbeat is \(\frac{6}{7}\) seconds, the heart rate \(f_1\) will be \[
f_1=\frac{60}{\frac{6}{7}}=70\,\text{beats per minute}.
\]
Similarly, for the function \(P_2\) the period will be \[
p_2=\frac{2\pi}{2\pi}=1\,\text{second}.
\] The heart rate is therefore \(f_2=60\) beats per minute.
Exercise 2. Draw the graphs of the functions from the first exercise. If you can, use appropriate software (for example GeoGebra) to draw the graphs. Use the blood pressure table to determine how the people in question are doing with their blood pressure.
Solution.
To plot the graph of the function, the \(x\)-axis will represent the time \(t\) in seconds. The \(y\)-axis will represent the blood pressure \(P\) in millimeters of mercury. Choose the units on the axes so that the graph looks clear. For example, a good choice is that one unit on the \(x\)-axis corresponds to one hundred units on the \(y\)-axis. In the solution in the figure, the ratio of units on the axes is \(1:125\). Function \[
P_1=25\cdot\sin\left(\frac{7\pi}{3}t\right)+105
\] oscillates around \(105\), with an amplitude equal to \(25\). The local maxima of the function will therefore have functional values \(105+25=130\) (systolic pressure). The local minima of the function will have functional values \(105-25=80\) (diastolic pressure).
The function \[ P_2=30\cdot\cos\left(2\pi t\right)+125 \] oscillates around the value \(125\), the amplitude is equal to \(30\). The local maxima of the function will therefore have the functional values \(125+30=155\) (systolic pressure). The local minima of the function will have the functional values \(125-30=95\) (diastolic pressure).
The function \(P_1\) is approximately a function of the pressure \(130\) over \(80\), according to the chart in the introduction it corresponds to the limit values between normal and high blood pressure (in some countries it is considered normal pressure, in some countries it is already the lower limit of high blood pressure). The function \(P_2\) corresponds to the pressure \(155\) over \(95\), this pressure is high.
Visualizing such graphs helps to understand changes in blood pressure and other periodic phenomena, which is necessary both in mathematical studies and in practical applications.
High blood pressure is a dangerous condition and a major risk factor for heart disease and stroke. A healthy lifestyle, such as a diet high in fruits and vegetables and low in sodium, as well as physical activity, can help prevent high blood pressure. A high value from a single measurement does not necessarily mean that you have high blood pressure, because blood pressure can be affected by many factors throughout the day, such as temperature, time of last meal, or stress.
By plotting the functions \(P_1\) and \(P_2\), we can see at a glance what the differences are between them. Sometimes, however, two functions given by different rules at first glance can have the same graph. For example, would you recognize at a glance that this is the case for the following two functions? \[ y=\sin\frac{3x}{5},\qquad y=\cos\left(\frac{3x}{5}-\frac{\pi}{2}\right) \] These functions have the same graph and the size of their period is \[ \frac{2\pi}{\frac{3}{5}}=\frac{10\pi}{3}. \]
But be careful, the graph of the function \(y=\cos\left(\frac{3x}{5}-\frac{\pi}{2}\right)\) is not shifted in the direction of the \(x\) axis by \(\frac{\pi}{2}\) compared to the graph of the function \(y=\cos\frac{3x}{5}\), as it might seem at first glance from the function formula, but by a quarter of the period of this function. We can see this if we modify the formula appropriately: \[ y=\cos\left(\frac{3x}{5}-\frac{\pi}{2}\right)=\cos\left(\frac{3}{5}\left(x-\frac{5\pi}{6}\right)\right) \]
To compare two such functions, it would be better if we could convert one to the other. This is exactly what the following exercise will deal with.
Exercise 3. Express the function \(P_1\) from Problem 1. using the cosine function instead of the sine function.
Solution. For the functions \(\sin x\) and \(\cos x\) in the basic form, \[ \sin x=\cos\left(x-\frac{\pi}{2}\right), \] where \(\frac{\pi}{2}\) is a quarter of the period. The period of the function \(P_1\) is \(p_1=\frac{6}{7}\), a quarter of the period is \[ \frac{p_1}{4}=\frac{6}{28}=\frac{3}{14}. \] Therefore, \[ \sin\left(\frac{7\pi}{3}t\right)=\cos\left(\frac{7\pi}{3}\left(t-\frac{3}{14}\right)\right) \] and the function \(P_1\) can be expressed as follows \[ P_1=25\cdot\cos\left(\frac{7\pi}{3}t-\frac{1}{2}\pi\right)+105. \]
In the previous problem, we could also interchange the functions and express the function \(P_2\) using the sine function.
Exercise 4. Find the formula for a function that approximates the blood pressure function of a healthy person at rest. His heart rate is \(50\) beats per minute. The maximum blood pressure is \(110\,\text{mm}\,\text{Hg}\) and the minimum is \(70\,\text{mm}\,\text{Hg}\).
Solution. To approximate the blood pressure function, we use, for example, the sine function (the solution for the cosine function would be similar).
The amplitude of the function is \(\frac{110-70}{2}=20\) and the function oscillates around the value \(\frac{110+70}{2}=90\).
The period of the function is \[ p=\frac{60}{50}=\frac{6}{5}, \] i.e. one heartbeat lasts \(1{,}2\) seconds. From the relation \[ p=\frac{2\pi}{b} = 1{,}2 \] for the period \(p\) of the function we get \(b= \frac{5}{3}\pi\).
The value of \(c\) can be chosen arbitrarily, the simplest is to choose \(c = 0.\) Substituting the above values into the general form of the function we get \[ P(t) = 20\cdot\sin\left(\frac{5\pi}{3}t\right)+90. \]
This function approximately models the blood pressure of a person with the specified values as a function of time (in seconds).
Finally, let’s explain how pressure is actually measured in reality. One of the accurate methods is the so-called auscultation technique. This method uses a tonometer, consisting of a rubber cuff, an inflatable bag and a manometer (mechanical pressure gauge), and a stethoscope.
The rubber cuff of the tonometer is placed approximately halfway up the arm. The pressure in the cuff is increased so that it exceeds the pressure in the artery. This turns the cuff into an artificial obstacle to blood flow. By gradually and slowly reducing the pressure in the cuff, blood flow is restored at a certain point.
However, the pressure in the cuff initially causes deformation of the artery, which makes the flow of penetrating blood turbulent. The pressure value at which heart sounds begin to be heard in the stethoscope corresponds to the value of systolic blood pressure.
The sounds are audible as long as the pressure in the cuff is sufficient to deform the artery and thus maintain turbulent flow. Once the pressure in the cuff drops so low that it is no longer sufficient to deform the artery, the original blood flow is restored and the echoes cease to be audible. The pressure at the last audible echo corresponds to the value of the diastolic blood pressure.
It remains to be added that a more realistic expression of the blood pressure function is more demanding and requires sums of trigonometric functions with different periods.
The following figure shows a specific example of such a sum and the corresponding graph. At the same time, the figure shows a blood pressure measurement. The cuff deflation rate is approximately constant. The pressure in the cuff therefore decreases at a constant rate (again approximately) and is shown in the figure by a straight line.
Refining the course of the blood pressure function using sums of sines and cosines is already related to the so-called Fourier theorem which states that a continuous periodic function can be expressed as the sum of an infinite number of sine and cosine functions, where each of these terms has a certain amplitude and period.
This result was obtained in 1822 by the French mathematician Joseph Fourier as part of the solution to the heat conduction equation. This is a key concept for the analysis and understanding of any periodic phenomena. Fourier’s theorem is the basis of signal processing.
Approximation means an approximate but faithful expression of a number or function, but also of a physical law or natural phenomenon.↩︎