**Trigonometry and Medicine**

**What is trigonometry?**

Trigonometry is a branch of mathematics that studies the correlation between lengths and angles of triangles. “Trigonometry began as the computational component of geometry. For instance, one statement of plane geometry states that a triangle is determined by a side and two angles. In other words, given one side of a triangle and two angles in the triangle, then the other two sides and the remaining angle are determined. Trigonometry includes the methods for computing those other two sides. The remaining angle is easy to find since the sum of the three angles equals 180 degrees (usually written 180°)” (Short 3).

Sine Waves

The aspect of Trigonometry that contributes to medicine most greatly is sine waves (sinusoidal functions). A sine wave is a curve that describes repetition and is named after a sine function. It occurs often in pure and applied mathematics, as well as physics, engineering, signal processing and medicine (electrical pulse machines). Its most basic form is a function of time (t) (Sine wave- Trigonometry 3). Sinusoidal waves indicate amplitude, frequency, period, midline and phase shifts. The equation for Sine waves are f(x)= Asin(B(x-C))+D and f(x)= Acos(B(x-C))+D. In 1822, Joseph Fourier discovered that sinusoidal waves can be used as building blocks to describe periodic waveforms including square, triangle, and saw tooth waves (Sine Waves,4). Without Trigonometry and Sinusoidal Waves, there would be a deference in the advancements of medicine, which includes Cardiology, Neurology, and Pulmonology.

The History of Trigonometry

The history of trigonometry goes back to as early as the 2nd millennium BC in Egyptian mathematics (Rhind Mathematical Papyrus). The second society to use trigonometry was the Babylonians(known for discovering theorems and ratios). The development of modern trigonometry shifted during the western Age of Enlightenment (Issaac Newton and James Stirling)(Joyve, 2013) It was Leonhard Euler who incorporated complex numbers into trigonometry. Moreover, Colin Maclaurin in the 18th century was pivotal in the development of trigonometric series (Joyce). Without Trigonometry and Sinusoidal Waves there would be a deference in the advancements of Cardiology, Neurology, and Pulmonology.

Cardiology

“Cardiology is the study and treatment of disorders of the heart. It is a medical specialty which is involved in the care of all things associated with the heart and the arteries” (Doctor Nordist). The field includes medical diagnosis and treatment of congenital heart defects, coronary artery disease, heart failure, valvular heart disease, and electrophysiology. Physicians who specialize in this field of medicine are called cardiologists. Physicians who specialize in cardiac surgery are called cardiac surgeons (Doctor Nordist).

EKG/ECG

An ECG or EKG is a very important machine in Cardiology. An EKG (electrocardiography) records heart activity by electricity. Furthermore, an EKG shows the speed of a heartbeat, the regularity of a heartbeat, and the strength and timing of electrical signals as they pass through each part of your heart (Heart Disease Health Center 1). Doctors use EKGs to detect and study many heart problems, such as heart attacks arrhythmias, and heart failure. The test results also can suggest other disorders that affect heart functions ( Heart Disease Health Center 4). A Dutch Indies Inventor Willem Einthoven discovered the Electrocardiography machine in 1901 and received a Nobel peace prize for his work in 1924 (A (not so) brief history of electrocardiography 4). The Electrocardiography machine records a graph that represents Sine waves. The way it does so is because a heartbeat is repititious it sets a tone for a cycle. The cycle has a amplitude and a period at all times and can have vertical and horizontal shift. The x-axis of the graph is time in milliseconds and the y-axis is represented by the voltage (amplitude). “Each1-mm-division on the horizontal axis is 40 ms; each 5-mm-division is 200 ms. Two 5-mm-divisions on the vertical axis are calibrated to represent 1 mV.

EEG

Breathing Rhythms

Pulmonolgist often track the rhythms of lung capacity. The reason is because the respiratory system function as a filter for air. When a person breathes with lungs, cells are supplied oxygen and removes carbon dioxide. In normal breathing patterns there is a rhythmic inhaling and exhaling. In result trigonometric equations can be applied to the respiration rhythm.

Neurology

“Neurology is the study of the nervous system and the brain, neurology deals with the diagnosis and treatment of diseases involving peripheral and the central nervous system” (Doctor Nordist). Doctors who specialize in Neurology are Neurosurgeons and Neurologist. A machine used in neurology often is a EEG machine (electroencephalography). A EEG measures voltage fluctuations resulting from ionic current flows within the neurons of the brain (electrical activities).

Pulumonology

“Pulmonolgy is a medical specialty that deals with the respitory tract. Pulmonolgy can also be classified under internal medicine or intensive care medicine. Doctors who are in this field are called pulmonologist and are trained to handle asthma, tuberculosis, pneumonia, and etc. Pulmonologist mainly handle paients under life support” (Doctor Nordist). Pulmonolgist often track the rhythms of lung capacity.

A German neurologist Hans Berger invented the EEG in 1929 and had an alpha wave named after him (Berger's Wave) (Berger, 24). EEG is built upon the system of sine waves. In a EEG there are different types of brain waves which include alpha, beta, theta, delta and gamma.

Pulmonology

“Pulmonology is a medical specialty that deals with the respiratory tract. Pulmonology can also be classified under internal medicine or intensive care medicine. Doctors who are in this field are called pulmonologist and are trained to handle asthma, tuberculosis, pneumonia, and etc. Pulmonologist mainly handle patients under life support” (Doctor Nordist). Pulmonologist often track the rhythms of lung capacity. The reason is because the respiratory system functions as a filter for air. When a person breathes through their lungs, cells supply oxygen and remove carbon dioxide. In normal breathing, there is a rhythmic inhaling and exhaling. As a result, trigonometric equations can be applied to the respiration rhythm.

In conclusion, trigonometry (Sinusoidal Waves) have contributed to various fields of medicine and testing. Without Sinusoidal waves, humanity would not have electrocardiography, electroencephalogram, and respiratory rhythms that assist in medical procedures. A world without electrocardiography would mean that we would not have a way of checking the heart's electrical activities. In addition, ECG test shines light on unexplained pains and discomfort that a patient may experience at any given time. Without a ECG test, it would be hard to diagnose chest pains, heart attacks, inflammation of the sac surrounding the heart, or angina. Furthermore, the ECG testing also determines the efficiency of medicine, the health of the heart, and the maintenance of mechanical implant devices. Moreover, a world without EEG testing would mean people would have no way of knowing if they have brain diseases, tumors, and chemistry changes and head infections. Parallel to ECG testing, EEG testing assist in determining unexplained afflictions such as confusion, trouble sleeping, and periods of unconsciousness (EEG 3). Furthermore, without the ability to calculate respiratory rhythms it would be hard to monitor people on life support in the aspect of breathing. Other aspects of respiration would be difficulty in tracking someone with asthma or pneumonia.

Real life application

A real world application showing an EKG as a Sine graph consists of the following. A person with a case of Hyperkalaemia is monitored on a EKG, The graph shows that he has a RR interval of 32 seconds (period). The graph also shows a amplitude of 1.23 volts of electricity, a midline of 2.4 volts, and a horizontal shift of 5 seconds. The trigonometric function would be f(x)=1.23sine[pi/16(t-5)]+2.4.

Application

If a doctor wanted to look at Delta waves in a child that was 8 months. The average test could read a amplitude of 120 volts, a polyspike (vertical shift) of 43,and a sharp wave (horizontal shift) of 90 milliseconds. The function would read f(x)=120sin[3(t+90)]+43

When a doctor is studying a person who is relaxed and has a constant rate of breathe, their average lung capacity is 5 seconds, amplitude of 250mL, tidal volume of 500mL, and vertical shift of 250mL. Since the following graph is a real world example of a sine wave the equation for it is: y=(A/2)sin(2pi/p(t))+S(y).