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Law of sines, cosines, tangents

Trigonometry is a branch of mathematics devoted to triangles, which allows you to find their unknown angles and faces from known values. For example, the angle along the length of the leg and hypotenuse, or the length of the hypotenuse according to the known angle and leg.

There are unique functions for calculations in trigonometry: sine, cosine, tangent, cotangent, secant and cosecant. They are often used in related sciences and disciplines, for example, in astronomy, geodesy, and architecture.

Trigonometry around us

Trigonometry is included in the general education curriculum and is one of the fundamental sections of mathematics. Today, with its help, geographic coordinates are found, ship routes are laid, the trajectories of celestial bodies are calculated, programs and statistical reports are compiled. This mathematical section is most in demand:

in astronomy;
in geography;
in navigation;
in architecture;
in optics;
in acoustics;
in economics (for the analysis of financial markets);
in probability theory;
in biology and medicine;
in electronics and programming.

Today even such seemingly abstract branches as pharmacology, cryptology, seismology, phonetics and crystallography cannot do without trigonometry. Trigonometric functions are used in computed tomography and ultrasound, to describe light and sound waves, in the construction of buildings and structures.

History of trigonometry

The first trigonometric tables were used in his writings by the ancient Greek scientist Hipparchus of Nicaea in 180-125 BC. Then they were purely applied in nature and were used only for astronomical calculations. There were no trigonometric functions (sine, cosine, and so on) in the tables of Hipparchus, but there was a division of the circle into 360 degrees and the measurement of its arcs using chords. For example, the modern sine was then known as "half a chord", to which a perpendicular was drawn from the center of the circle.

In the year 100 AD, the ancient Greek mathematician Menelaus of Alexandria, in his three-volume "Sphere" (Sphaericorum), presented several theorems that today can be fully considered "trigonometric". The first described the congruence of two spherical triangles, the second the sum of their angles (which is always greater than 180 degrees), and the third the "six magnitudes" rule, better known as the Menelaus theorem.

Roughly at the same time, from AD 90 to 160, the astronomer Claudius Ptolemy published the most significant trigonometric treatise of antiquity, Almagest, consisting of 13 books. The key to it was a theorem describing the ratio of diagonals and opposite sides of a convex quadrilateral inscribed in a circle. According to Ptolemy's theorem, the product of the second is always equal to the sum of the products of the first. Based on it, 4 difference formulas for sine and cosine were subsequently developed, as well as the half-angle formula α / 2.

Indian Studies

The "chordal" form of describing trigonometric functions, which arose in ancient Greece before our era, was common in Europe and Asia until the Middle Ages. And only in the 16th century in India they were replaced by the modern sine and cosine: with the Latin designations sin and cos, respectively. It was in India that the fundamental trigonometric ratios were developed: sin²α + cos²α = 1, sinα = cos(90° − α), sin(α + β) = sinα ⋅ cosβ + cosα ⋅ sinβ and others.

The main purpose of trigonometry in medieval India was to find ultra-precise numbers, primarily for astronomical research. This can be judged from the scientific treatises of Bhaskara and Aryabhata, including the scientific work Surya Siddhanta. The Indian astronomer Nilakanta Somayaji for the first time in history decomposed the arctangent into an infinite power series, and subsequently the sine and cosine were decomposed into series.

In Europe, the same results came only in the next, XVII century. The series for sin and cos were derived by Isaac Newton in 1666, and for the arc tangent in 1671 by Gottfried Wilhelm Leibniz. In the 18th century, scientists were engaged in trigonometric studies both in Europe and in the countries of the Near / Middle East. After Muslim scientific works were translated into Latin and English in the 19th century, they became the property of first European and then world science, made it possible to combine and systematize all knowledge related to trigonometry.

Summing up, we can say that today trigonometry is an indispensable discipline not only for the natural sciences, but also for information technology. It has long ceased to be an applied branch of mathematics, and consists of several large subsections, including spherical trigonometry and goniometry. The first considers the properties of angles between great circles on a sphere, and the second deals with methods for measuring angles and the ratio of trigonometric functions to each other.

Sine, cosine, tangent formulas

Trigonometry is primarily about finding corners and edges in right triangles, as well as in more complex, polyhedral shapes. Knowing two quantities (an angle and a face or two faces), you can almost always find the third one using special trigonometric functions and formulas.

Trigonometric functions

There are only two direct functions in trigonometry: sine (sin) and cosine (cos). The first is equal to the ratio of the opposite leg to the hypotenuse, and the second is equal to the adjacent. In both cases, we mean the acute angle of a right triangle, which is always less than 90 degrees. In higher mathematics, sin and cos can also be applied to complex and real numbers.

All other trigonometric functions are derivatives of sine and cosine. There are only four of them:

Tangent (tg) - the ratio of the opposite leg to the adjacent one - tgx = sinx / cosx.
Cotangent (ctg) - the ratio of the adjacent leg to the opposite one - ctgx = cosx / sinx.
Second (sec) — the ratio of the hypotenuse to the adjacent leg — secx = 1 / cosx.
Cosecant (cosec) - the ratio of the hypotenuse to the opposite leg - cosecx = 1 / sinx.

An alternative notation used in English-speaking countries is as follows: tangent - tan, cotangent - cot, cosecant - csc. They are indicated in the scientific literature, on push-button engineering calculators, in electronic applications.

Trigonometric formulas

Mathematicians of European and Asian countries have been researching and improving trigonometric functions for many centuries, and have identified a number of patterns inherent in them in addition, subtraction, multiplication and other mathematical operations. Today, the entire basic course of trigonometry, which is part of the school curriculum, is based on this, namely, the ability to reduce and transform functions using existing axioms and theorems.

Simple identities

Even in medieval India, the simplest identities applicable to direct and derivative trigonometric functions were revealed. In their finished (modern) form, they look like this:

sin²α + cos²α = 1.
1 + tg²α = sec²α.
1 + ctg²α = cosec²α.
tgα ⋅ ctgα = 1.

The above formulas are valid for any values of the argument (α). If we introduce the constraint that α is greater than 0 and less than π/2, the list of formulas increases several times. The main ones include the following:

sinα = √(1 − cos²α).
cosα = √(1 − sin²α).
tgα = sinα / √(1 − sin²α).
ctg = cosα / √(1 − cos²α).
sec = 1 / cosα.
cosec = 1 / sinα.

There are 5 valid identities for each of the 6 functions (30 in total). All of them are listed in the table and can be used to solve and simplify trigonometric equations with one unknown (α).

Addition and subtraction

The sums and differences of two angles (α and β) also have their own patterns. Using trigonometric formulas, they can be represented as follows:

sin(α + β) = sinα ⋅ cosβ + sinβ ⋅ cosα.
cos(α + β) = cosα ⋅ cosβ + sinα ⋅ sinβ.
tg(α + β) = (tgα + tgβ) / (1 − tgα ⋅ tgβ).
ctg(α + β) = (ctgα ⋅ ctgβ − 1) / (ctgα + ctgβ).

These formulas apply to subtraction as well. If the signs on the right side of the equal sign change, then they also change on the left side. In the case of the tangent, it will look like this: tg(α − β) = (tgα − tgβ) / (1 + tgα ⋅ tgβ).

Multiplication

The trigonometric functions of two angles (α and β) can also be multiplied together using the existing formulas:

sinα ⋅ sinβ = (cos(α − β) − cos(α + β)) / 2.
sinα ⋅ cosβ = (sin(α − β) + sin(α + β)) / 2.
cosα ⋅ cosβ = (cos(α − β) + cos(α + β)) / 2.
tgα ⋅ tgβ = (cos(α − β) − cos(α + β)) / (cos(α − β) + cos(α + β)).
tgα ⋅ ctgβ = (sin(α − β) + sin(α + β)) / (sin(α + β) − sin(α − β)).
ctgα ⋅ ctgβ = (cos(α − β) + cos(α + β)) / (cos(α − β) − cos(α + β)).

There are also formulas for raising trigonometric functions to a power, for universal substitution, for expanding into infinite products, for obtaining derivatives and antiderivatives. The length of formulas can vary from 2-3 to tens of characters, using integrals, products of polynomials, hyperbolic functions. They are not easy to calculate even with simple values of α and β, and if they are complex fractional values with many decimals, the calculations will require a lot of time and effort.

To simplify the calculations of trigonometric functions (and operations with them), today special online calculators are used. Numerical values are entered into them, after which the program calculates in a fraction of a second. Using such applications is even more convenient than engineering calculators, and they are available completely free of charge.