Trigonometry

 

The Origins of Trigonometric Functions (sine, cosine, tangent, cotangent, secant, and cosecant)

A brief history and etymology of trigonometric functions from ancient to modern times

Sunny Labh

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Trigonometry is a rich and fascinating branch of mathematics that has a long and diverse history. It has been influenced by many cultures and civilizations, and has contributed to many fields of knowledge and human endeavor. Trigonometric functions are mathematical functions that relate the angles and sides of right triangles. They are widely used in geometry, astronomy, engineering, and many other fields of science and mathematics. The most common trigonometric functions are sine, cosine, and tangent, and their reciprocals are cosecant, secant, and cotangent. But where did these functions come from? How did they get their names and symbols? In this article, I’ll try to explain the history and etymology of trigonometric functions, from ancient times to modern times.

The earliest traces of trigonometry can be found in the ancient civilizations of Egypt and Babylon, where they used theorems on the ratios of the sides of similar triangles to measure distances and angles. However, they did not have a concept of an angle measure, so they only studied the lengths of the sides of triangles. The first systematic study of trigonometric functions began in Hellenistic mathematics, around the 3rd century BC, by Greek astronomers and mathematicians such as Hipparchus, Menelaus, and Ptolemy. They used the concept of a chord, which is a line segment that joins two points on a circle, to define trigonometric functions. They also developed tables of chords for various angles, which allowed them to calculate the positions and movements of celestial bodies.

Illustration of Hipparchus observing the night sky. Created by Bruno H. Brugel. Public Domain Image. Source: Commons

The word “trigonometry” comes from the Greek words “trigonon” (triangle) and “metron” (measure), meaning the measurement of triangles. The word “chord” comes from the Latin word “chorda”, meaning string or rope.

The next major development in trigonometry occurred in India, where mathematicians such as Aryabhata, Brahmagupta, Bhaskara, and Varahamihira expanded the scope and accuracy of trigonometric calculations. They introduced the concept of a sine, which is half a chord of a unit circle (a circle with radius 1), and discovered many important formulas and identities involving sines. They also developed tables of sines and versines (the complement of sines) for various angles, which improved the precision and efficiency of astronomical computations.

Image by the author

The word “sine” comes from the Latin word “sinus”, meaning curve or fold. This is a mistranslation of the Arabic word “jiba”, which is a transliteration of the Sanskrit word “jiva”, meaning bowstring or chord. The Arabic word was also sometimes written as “jaib”, which means bosom or bay, and this may have confused the Latin translators.

The transmission of trigonometry from India to the Islamic world was facilitated by the expansion of trade and cultural exchange between the 8th and 10th centuries AD. Muslim mathematicians such as Al-Khwarizmi, Al-Battani, Al-Karaji, Al-Biruni, Al-Nasawi, Al-Tusi, and Al-Wafa made significant contributions to trigonometry by introducing new concepts and methods. They defined six trigonometric functions: sine, cosine, tangent, cotangent, secant, and cosecant. They also used the concept of a half-angle to simplify calculations and derived many new formulas and identities involving trigonometric functions. They applied trigonometry to various fields such as astronomy, geography, navigation, surveying, optics, and music theory.

graphics by the author

Etymology

The words “cosine”, “tangent”, “cotangent”, “secant”, and “cosecant” come from Latin translations or adaptations of Arabic terms. The word “cosine” is an abbreviation of “cosinus”, meaning sine of the complement (the angle that is subtracted from 90 degrees). The word “tangent” comes from “tangens”, meaning touching or extending (the line that touches the circle at one point). The word “cotangent” is an abbreviation of “cotangens”, meaning tangent of the complement. The word “secant” comes from “secans”, meaning cutting or intersecting (the line that cuts the circle at two points). The word “cosecant” is an abbreviation of “cosecans”, meaning secant of the complement.

The spread of trigonometry from the Islamic world to Europe was stimulated by the Crusades and the Renaissance in the 12th to 16th centuries AD. European scholars translated Arabic texts into Latin and learned from Muslim scientists and philosophers. They also added their own discoveries and innovations to trigonometry. Some notable figures in this period include Fibonacci, Regiomontanus, Copernicus, Kepler, Napier, Briggs, and Viète. They developed new methods of calculating trigonometric functions, such as logarithms, decimal fractions, and infinite series. They also applied trigonometry to new areas such as physics, mechanics, and astronomy.

The modern notation and symbols for trigonometric functions were established in the 17th and 18th centuries AD by mathematicians such as Oughtred, Descartes, Newton, Leibniz, Euler, and Gauss. They used the abbreviations “sin”, “cos”, “tan”, “cot”, “sec”, and “csc” for the six trigonometric functions, and used parentheses to indicate function arguments. They also used exponents to indicate powers of functions, and used inverse functions to solve equations involving trigonometric functions. They extended the domain of trigonometric functions from real numbers to complex numbers, and generalized trigonometry to non-right triangles and non-Euclidean geometries. Trigonometry is a rich and fascinating branch of mathematics that has a long and diverse history. It has been influenced by many cultures and civilizations, and has contributed to many fields of knowledge and human endeavor. It continues to be an active and relevant subject in modern mathematics and science, as well as in art, music, and culture.

The sine curve. Image by the author.

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SUGAR

 

Sweets, like these candies at La Boqueria Market in Barcelona, release dopamine in our brains on levels similar to nicotine and alcohol.

PHOTOGRAPH COURTESY GETTY IMAGES

• SCIENCE

How sugar and fat affect your brain

Much of our food is increasingly manufactured to be irresistible to us. Experts say this trend has long-term health consequences.

BYALLIE YANG

PUBLISHED DECEMBER 28, 2022

• 9 MIN READ

We often think of smoking and drinking when it comes to addiction—but there’s another compulsion affecting as many as 14 percent of adults and even 12 percent of kids: food addiction.

Indulgent dishes enticing us with fat and sugar can feel impossible to avoid, especially during the holidays. Experts confirm it’s more than a feeling: Half a century of food trends has created an environment where more than half the food consumed by American adults is ultra-processed, often optimized to hit the body’s fat and sugar sensors to release dopamine. 

These processed food products capitalize on our biology to keep us reaching for more. “We don't realize that these are really killing people on par with what we're seeing with things like alcohol and tobacco, leading to preventable deaths,” says Ashley Gearhardt, an associate professor of psychology at the University of Michigan and a member of a research team that assessed the latest figures for the prevalence of food addiction in March 2022.

Experts are rewriting what we know about food addiction and asking new questions about what we can do to curb it—and save lives.

What food does to our brains

Food affects our brains in many complex ways, and one particularly important response is the release of dopamine, a neurotransmitter. Like addictive drugs, eating food releases dopamine. Contrary to popular belief, dopamine doesn't increase pleasure. It encourages us to repeat behaviors that help us survive—like eating nutritious food and reproducing. The more dopamine that’s released, the more likely we’ll repeat that behavior.

When we eat fat and sugar, sensors in the mouth send a message to release dopamine in the striatum, a section of the brain associated with movement and rewarding behavior. But that oral sensory process is only part of the story, says Alexandra DiFeliceantonio, an assistant professor at Virginia Tech’s Fralin Biomedical Research Institute. There’s also a secondary sensor in the gut that registers fat and sugar, signaling the brain to release dopamine in the same region. 

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Although researchers are still mapping out how exactly the presence of sugar is signaled from the gut to the brain, the way that fat is signaled from the gut to the brain has been well documented. When fat is detected in the upper intestine, the message is carried up the vagus nerve (which controls several unconscious functions like digestion and breathing) through the hindbrain to the striatum.

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Foods rich in fat and sugar can increase dopamine in the striatum as much as 200 percent above normal levels—a similar bump to what’s observed with nicotine and alcohol, the two most common addictions in the U.S. Specifically, one study found sugar increased dopamine levels by 135 to 140 percent, and fat increased them by 160 percent in another study, although it takes longer to kick in. Other drugs work very differently—cocaine can triple normal dopamine levels while methamphetamine can multiply normal dopamine levels 10-fold.

How the food we eat has changed

As we learn more about how food affects our brains, it has become increasingly manufactured to be irresistible to us. Our bodies are inundated with foods that have higher concentrations of certain nutrients, like fat and sugar, and more combinations of nutrients than ever before. These are combined with sensory properties—like a pleasingly smooth and velvety ice cream—that make eating more enjoyable than ever.

Traditionally, humans made food with whole foods: for example, pie crusts were made from flour and butter. In contrast, industrially processed foods are composed of substances extracted from foods, like starches and hydrogenated fats. Additives like artificial flavors, emulsifiers (which keep oil and water mixed together), and stabilizers (which preserve the structure or texture of food) make food more appealing—but ultimately to our own detriment.

Experts like DiFeliceantonio believe we should make the distinction between highly processed foods and those made from scratch. Being aware of those differences is the first step in avoiding a long list of diet-related health issues.

“We've been eating homemade versions of cakes, cookies, and pizzas for a very long time. But, it wasn't until the rise in production of ultra-processed foods in the 1980s that we've seen this increase in diet-related mortality and disease,” DiFeliceantonio says.

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Highly processed foods can qualify as clinically addictive, both Gearhardt and DiFeliceantonio argue. According to what’s known as the rate hypothesis, the faster something affects your brain, the more addictive that substance will be. Many processed foods are essentially pre-digested to maximize the speed of dopamine release.

Finally, it’d be impossible to extricate social and psychological forces from the equation. Processed foods have been accessible, affordable, and aggressively advertised for generations. That perfect storm has created generations of people who know processed foods aren’t healthy but are still compulsively drawn to them.

“The cues that surround these foods start to take on a life of their own,” says Gearhardt. “When you see a fast food sign or a vending machine, that has such power and drive for us that even if you're not hungry, or even if your doctor just told you have diabetes, you might want to have these processed foods you know aren't good for you. They're everywhere; we're constantly on the defensive against doughnuts at the morning meeting, and the late-night advertisement for pizza.”

How are perspectives changing? What are the questions left unanswered?

In recent years, experts have begun to ask new questions about food addiction as some of their early assumptions have proven false.

Take tolerance and withdrawal for example. These were once considered major elements of addiction. It was once believed people with food addiction continued to eat compulsively in order to avoid withdrawal, the unpleasant physical and mental repercussions (like anxiety, nausea, and headaches) that appear when a person decreases or stops using a substance.

“That's actually not true,” DiFeliceantonio says. “Most theories of drug addiction have a lot more to do with habitual use, or with an intense craving. That's what’s sustaining drug use.”

Tolerance is almost the opposite of withdrawal—the consequences of continuing to use a substance. As a person’s tolerance of a substance grows, they need to consume increasing amounts to get the same effect. In the case of food, the dopamine deficit hypothesis posits that if we eat something and don’t get enough pleasure from it, we’ll eat more until we feel good.

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“I have some trouble with that hypothesis, because everything we do releases dopamine. So eating broccoli, because it is delivering nutrients to the gut, releases dopamine,” DiFeliceantonio says. “People don't do things that they only kind of like, such as eating broccoli, a lot just to get more dopamine.” She adds there’s also no indication there’s a threshold to reach to earn that dopamine reward.

As research progresses, scientists are left with more questions than answers about how our bodies become addicted to food. We know that dopamine doesn’t tell the full story, because it’s not what makes eating food pleasurable. Researchers have found evidence that might actually have a different cause: A 2012 study showed that eating food stimulates our opioid receptors, which increase feelings of pleasure. But scientists know very little about how the process works, because it’s difficult to measure opioid levels in a living organism.

Some experts suspect a sensor in the upper intestine may play a role in our food likes and dislikes. Others wonder whether there may be something at play in the hypothalamus, a critical part of the brain that regulates everything from your body temperature to your sense of hunger.

Researchers also want to learn which combinations of nutrients trigger different levels of dopamine release. Studying humans, unfortunately, requires expensive scans and a dose of radiation. “You can’t scan the same person 20 times with all different tastes and combinations and things, so we're really more limited in what we can do,” DiFeliceantonio says.

As for a solution, Gearhardt says the answer is clear—but far from easy. We can look to the major societal changes that were imposed to limit smoking—making cigarettes less affordable and less widely marketed—and do the same with addictive food, she says.

There are other ways to fight food addiction, too. 

“Don't hate yourself for not being able to avoid addictive foods because it isn't easy. It's using our biology against us,” she says. Learn what makes you turn to these foods, whether it’s certain emotions, places or even a time of day. “Just try to be aware of that so you can prepare yourself to have alternative ways of coping or strategizing in those moments of temptation.”

A legalidade

 

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A legalidade em imagens

25 de junho de 2011 | Categorias: 1

Para marcar os 50 anos da Legalidade, o PDT editou sete postais com fotos do movimento comandado pelo então governador Leonel Brizola, em defesa da posse de João Goulart na presidência.

Confira abaixo as imagens, que integram o acervo do Museu de Comunicação Hipólito José da Costa e retratam os 13 dias da mobilização comandada pelo líder trabalhista desde a renúncia do presidente Jânio Quadros, em 25 de agosto de 1961, até a posse de Jango, em 7 de setembro.

Ouça também o hino da legalidade: