Trigonometry

Trigonometry (from other Greek: τρίγωνον “ triangle ” and μετρέω “measure”, that is, the measurement of triangles ) is a branch of mathematics that studies trigonometric functions and their use in geometry ^[1] . This term first appeared in 1595 as the name of the book of the German mathematician Bartolomeus Pitiskus (1561-1613), and science itself was used in ancient times for calculations in astronomy, architecture and geodesy (a science that studies the size and shape of the Earth).

Trigonometric calculations are used in almost all areas of geometry , physics and engineering . Of great importance is the technique of triangulation , which allows you to measure distances to nearby stars in astronomy , between landmarks in geography , to control satellite navigation systems. It should also be noted the use of trigonometry in areas such as music theory , acoustics , optics , financial market analysis, electronics , probability theory , statistics , biology , medicine (including ultrasound (ultrasound) and computed tomography ), pharmaceuticals , chemistry , number theory ( and, as a consequence, cryptography ), seismology , meteorology , oceanography , cartography , many areas of physics , topography and geodesy , architecture , phonetics , economics , electronic engineering , mechanical engineering , computer graphics , crystal Fargo .

Content

History

Ancient Greece

The first trigonometric tables were apparently compiled by Hipparchus , who is now known as the “father of trigonometry” ^[2] .

Ancient Greek mathematicians in their constructions related to the measurement of circular arcs used the technique of chords. The perpendicular to the chord, lowered from the center of the circle, halves the arc and the chord resting on it. Half of the chord divided in half is the sine of the half angle, and therefore the sine function is also known as the "half of the chord." Due to this dependence, a significant number of trigonometric identities and theorems known today were also known to ancient Greek mathematicians, but in equivalent chord form. Although there is no trigonometry in the strict sense of the word in the works of Euclid and Archimedes, their theorems are presented in geometric form equivalent to specific trigonometric formulas. Archimedes' theorem for dividing chords is equivalent to the formulas for the sines of the sum and difference of angles. To compensate for the absence of a table of chords, mathematicians from the times of Aristarchus sometimes used the well-known theorem, in the modern notation - sinα / sinβ <α / β <tgα / tgβ, where 0 ° <β <α <90 °, together with other theorems.

The first trigonometric tables were probably compiled by Hipparchus of Nicaea (180-125 years BC). Hipparchus was the first to tabulate the corresponding values of arcs and chords for a series of angles. The systematic use of a full circle of 360 ° was established mainly due to Hipparchus and his chord table. Perhaps Hipparchus took the idea of such a division from Gipsicle , which had previously divided the day into 360 parts, although Babylonian astronomers could also offer such a division of the day.

Menelaus of Alexandria (100 AD) wrote The Sphere in three books. In the first book, he presented the basics for spherical triangles , similar to the first book of Euclid ’s beginning on flat triangles. He presented a theorem for which Euclid has no analogue that two spherical triangles are congruent if the corresponding angles are equal, but he did not distinguish between congruent and symmetric spherical triangles. His other theorem states that the sum of the angles of a spherical triangle is always greater than 180 °. The second book, Spherics, applies spherical geometry to astronomy. The third book contains the " Menelaus theorem ", also known as the "rule of six quantities."

Later, Claudius Ptolemy (90 - 168 AD) in the Almagest expanded the Hipparchian Chords in a Circle. Thirteen books of the Almagest is the most significant trigonometric work of all antiquity. The theorem that was central to the calculation of Ptolemy’s chords is also known today as Ptolemy’s theorem , which says that the sum of the products of the opposite sides of a convex inscribed quadrilateral is equal to the product of the diagonals. A separate case of Ptolemy’s theorem appeared as the 93rd sentence of Euclid’s “Data”.

Ptolemy's theorem entails the equivalence of the four sum and difference formulas for sine and cosine. Ptolemy later deduced the half-angle formula. Ptolemy used these results to create his trigonometric tables, although perhaps these tables were derived from the work of Hipparchus.

Medieval India

Replacing chords with sines was the main achievement of medieval India. Such a replacement allowed the introduction of various functions associated with the sides and corners of a right triangle. Thus, in India, trigonometry was laid as the doctrine of trigonometric quantities.

Indian scientists used various trigonometric relationships, including those that are expressed in modern form as

$\sin ^{2}\alpha +\cos ^{2}\alpha =1,$ ${\ displaystyle \ sin ^ {2} \ alpha + \ cos ^ {2} \ alpha = 1,}$

$\sin \alpha =\cos(90^{\circ }-\alpha ),$ ${\ displaystyle \ sin \ alpha = \ cos (90 ^ {\ circ} - \ alpha),}$

$\sin(\alpha \pm \beta )=\sin \alpha \cos \beta \pm \cos \alpha \sin \beta .$ ${\ displaystyle \ sin (\ alpha \ pm \ beta) = \ sin \ alpha \ cos \ beta \ pm \ cos \ alpha \ sin \ beta.}$

Indians also knew formulas for multiple angles $\sin n\alpha ,\qquad \cos n\alpha ,$ ${\ displaystyle \ sin n \ alpha, \ qquad \ cos n \ alpha,}$ Where $n=2,3,4,5.$ ${\ displaystyle n = 2,3,4,5.}$

Trigonometry is necessary for astronomical calculations, which are made out in the form of tables. The first table of sinuses is found in Surya Siddhanta and Ariabhata . Later, scientists compiled more detailed tables: for example, Bhaskara gives a table of sines through 1 °.

South Indian mathematicians in the sixteenth century made great strides in the summation of infinite number series. Apparently, they were engaged in these studies when they were looking for ways to calculate more accurate values of the number π . Nilacantha verbally brings the rules of decomposing the arc tangent into an infinite power series. And in the anonymous treatise “ ” (“Technique of Computing”), the rules for the expansion of sine and cosine in infinite power series are given. I must say that in Europe, similar results came only in the 17-18 centuries. So, the series for sine and cosine was derived by Isaac Newton around 1666, and the series of arctangent was found by J. Gregory in 1671 and G.V. Leibniz in 1673.

Since the VIII century, scientists from the countries of the Near and Middle East developed the trigonometry of their predecessors. In the middle of the 9th century, the Central Asian scholar al-Khwarizmi wrote the essay “ On the Indian Account ”. After the treatises of Muslim scholars were translated into Latin, many ideas of Greek, Indian and Muslim mathematicians became the property of European and then world science.

Definition of trigonometric functions

Trigonometric functions of the angle θ inside the unit circle

Initially, trigonometric functions were associated with aspect ratios in a right-angled triangle . Their only argument is the angle (one of the sharp corners of this triangle).

Sinus is the ratio of the opposite side to hypotenuse .
Cosine is the ratio of the adjacent cathetus to hypotenuse.
The tangent is the ratio of the opposite side to the adjacent.
Cotangent - the ratio of the adjacent side to the opposite.
Secant - the ratio of the hypotenuse to the adjacent leg.
Cosecans - the ratio of the hypotenuse to the opposite leg.

These definitions allow you to calculate the values of the functions for acute angles, that is, from 0 ° to 90 ° (from 0 to $\pi \over 2$ ${\ displaystyle \ pi \ over 2}$ radian). In the 18th century, Leonhard Euler gave modern, more general definitions, expanding the scope of these functions to the entire numerical axis . Consider a circle of unit radius in a rectangular coordinate system (see figure) and set aside the angle from the horizontal axis $\theta$ ${\ displaystyle \ theta}$ (if the angle is positive, then put it off counterclockwise, otherwise clockwise). The intersection point of the constructed side of the angle with the circle is denoted by A. Then:

Sine of angle $\theta$ ${\ displaystyle \ theta}$ defined as the ordinate of point A.
The cosine is the abscissa of point A.
The tangent is the ratio of sine to cosine.
Cotangent is the ratio of cosine to sine (i.e., the reciprocal of the tangent).
Secant is the reciprocal of the cosine.
Cosecance is the reciprocal of the sine.

For acute angles, the new definitions coincide with the previous ones.

A purely analytical definition of these functions is also possible, which is not related to geometry and represents each function by its expansion into an infinite series.

Sine Function Properties

Sinus

The domain of the function is the set of all real numbers: $D(y)=R$ ${\ displaystyle D (y) = R}$ .
The set of values is the interval [−1; one]: $E(y)$ ${\ displaystyle E (y)}$ = [−1; 1].
Function $y=\sin \left(\alpha \right)$ ${\ displaystyle y = \ sin \ left (\ alpha \ right)}$ is odd: $\sin \left(-\alpha \right)=-\sin \alpha$ ${\ displaystyle \ sin \ left (- \ alpha \ right) = - \ sin \ alpha}$ .
The function is periodic, the smallest positive period is $2\pi$ ${\ displaystyle 2 \ pi}$ : $\sin \left(\alpha +2\pi \right)=\sin \left(\alpha \right)$ ${\ displaystyle \ sin \ left (\ alpha +2 \ pi \ right) = \ sin \ left (\ alpha \ right)}$ .
The function graph crosses the axis Ox at $\alpha =\pi n\,,n\in \mathbb {Z}$ ${\ displaystyle \ alpha = \ pi n \ ,, n \ in \ mathbb {Z}}$ .
Permanent intervals: $y>0$ ${\ displaystyle y> 0}$ at $\left(2\pi n+0;\pi +2\pi n\right)\,,n\in \mathbb {Z}$ ${\ displaystyle \ left (2 \ pi n + 0; \ pi +2 \ pi n \ right) \ ,, n \ in \ mathbb {Z}}$ and $y<0$ ${\ displaystyle y <0}$ at $\left(\pi +2\pi n;2\pi +2\pi n\right)\,,n\in \mathbb {Z}$ ${\ displaystyle \ left (\ pi +2 \ pi n; 2 \ pi +2 \ pi n \ right) \ ,, n \ in \ mathbb {Z}}$ .
The function is continuous and has a derivative for any value of the argument: $(\sin \alpha )'=\cos \alpha$ ${\ displaystyle (\ sin \ alpha) '= \ cos \ alpha}$
Function $y=\sin \alpha$ ${\ displaystyle y = \ sin \ alpha}$ increases with $\alpha \in \left(-{\frac {\pi }{2}}+2\pi n;{\frac {\pi }{2}}+2\pi n\right)\,,n\in \mathbb {Z}$ ${\ displaystyle \ alpha \ in \ left (- {\ frac {\ pi} {2}} + 2 \ pi n; {\ frac {\ pi} {2}} + 2 \ pi n \ right) \ ,, n \ in \ mathbb {Z}}$ , and decreases with $\alpha \in \left({\frac {\pi }{2}}+2\pi n;3{\frac {\pi }{2}}+2\pi n\right)\,,n\in \mathbb {Z}$ ${\ displaystyle \ alpha \ in \ left ({\ frac {\ pi} {2}} + 2 \ pi n; 3 {\ frac {\ pi} {2}} + 2 \ pi n \ right) \ ,, n \ in \ mathbb {Z}}$ .
The function has a minimum at $\alpha =-{\frac {\pi }{2}}+2\pi n\,,n\in \mathbb {Z}$ ${\ displaystyle \ alpha = - {\ frac {\ pi} {2}} + 2 \ pi n \ ,, n \ in \ mathbb {Z}}$ and maximum at $\alpha ={\frac {\pi }{2}}+2\pi n\,,n\in \mathbb {Z}$ ${\ displaystyle \ alpha = {\ frac {\ pi} {2}} + 2 \ pi n \ ,, n \ in \ mathbb {Z}}$ .

Cosine Function Properties

Cosine

The domain of the function is the set of all real numbers: $D(y)=R$ ${\ displaystyle D (y) = R}$ .
The set of values is the interval [−1; one]: $E(y)$ ${\ displaystyle E (y)}$ = [−1; 1].
Function $y=\cos \left(\alpha \right)$ ${\ displaystyle y = \ cos \ left (\ alpha \ right)}$ is even: $\cos \left(-\alpha \right)=\cos \alpha$ ${\ displaystyle \ cos \ left (- \ alpha \ right) = \ cos \ alpha}$ .
The function is periodic, the smallest positive period is $2\pi$ ${\ displaystyle 2 \ pi}$ : $\cos \left(\alpha +2\pi \right)=\cos \left(\alpha \right)$ ${\ displaystyle \ cos \ left (\ alpha +2 \ pi \ right) = \ cos \ left (\ alpha \ right)}$ .
The function graph crosses the axis Ox at $\alpha ={\frac {\pi }{2}}+\pi n\,,n\in \mathbb {Z}$ ${\ displaystyle \ alpha = {\ frac {\ pi} {2}} + \ pi n \ ,, n \ in \ mathbb {Z}}$ .
Permanent intervals: $y>0$ ${\ displaystyle y> 0}$ at $\left(-{\frac {\pi }{2}}+2\pi n;{\frac {\pi }{2}}+2\pi n\right)\,,n\in \mathbb {Z}$ ${\ displaystyle \ left (- {\ frac {\ pi} {2}} + 2 \ pi n; {\ frac {\ pi} {2}} + 2 \ pi n \ right) \ ,, n \ in \ mathbb {Z}}$ and $y<0$ ${\ displaystyle y <0}$ at $\left({\frac {\pi }{2}}+2\pi n;3{\frac {\pi }{2}}+2\pi n\right)\,,n\in \mathbb {Z} .$ ${\ displaystyle \ left ({\ frac {\ pi} {2}} + 2 \ pi n; 3 {\ frac {\ pi} {2}} + 2 \ pi n \ right) \ ,, n \ in \ mathbb {Z}.}$
The function is continuous and has a derivative for any value of the argument: $(\cos \alpha )'=-\sin \alpha$ ${\ displaystyle (\ cos \ alpha) '= - \ sin \ alpha}$
Function $y=\cos \alpha$ ${\ displaystyle y = \ cos \ alpha}$ increases with $\alpha \in \left(-\pi +2\pi n;2\pi n\right)\,,n\in \mathbb {Z} ,$ ${\ displaystyle \ alpha \ in \ left (- \ pi +2 \ pi n; 2 \ pi n \ right) \ ,, n \ in \ mathbb {Z},}$ and decreases with $\alpha \in \left(2\pi n;\pi +2\pi n\right)\,,n\in \mathbb {Z} .$ ${\ displaystyle \ alpha \ in \ left (2 \ pi n; \ pi +2 \ pi n \ right) \ ,, n \ in \ mathbb {Z}.}$
The function has a minimum at $\alpha =\pi +2\pi n\,,n\in \mathbb {Z}$ ${\ displaystyle \ alpha = \ pi +2 \ pi n \ ,, n \ in \ mathbb {Z}}$ and maximum at $\alpha =2\pi n\,,n\in \mathbb {Z} .$ ${\ displaystyle \ alpha = 2 \ pi n \ ,, n \ in \ mathbb {Z}.}$

Tangent Function Properties

Tangent

The domain of the function is the set of all real numbers: $D(y)=R$ ${\ displaystyle D (y) = R}$ except numbers $\alpha ={\frac {\pi }{2}}+\pi n,n\in \mathbb {Z} \,.$ ${\ displaystyle \ alpha = {\ frac {\ pi} {2}} + \ pi n, n \ in \ mathbb {Z} \ ,.}$
The set of values is the set of all real numbers: $E(y)=R.$ ${\ displaystyle E (y) = R.}$
Function $y=\mathrm {tg} \left(\alpha \right)$ ${\ displaystyle y = \ mathrm {tg} \ left (\ alpha \ right)}$ is odd: $\mathrm {tg} \left(-\alpha \right)=-\mathrm {tg} \ \alpha$ ${\ displaystyle \ mathrm {tg} \ left (- \ alpha \ right) = - \ mathrm {tg} \ \ alpha}$ .
The function is periodic, the smallest positive period is $\pi$ ${\ displaystyle \ pi}$ : $\mathrm {tg} \left(\alpha +\pi \right)=\mathrm {tg} \left(\alpha \right)$ ${\ displaystyle \ mathrm {tg} \ left (\ alpha + \ pi \ right) = \ mathrm {tg} \ left (\ alpha \ right)}$ .
The function graph crosses the axis Ox at $\alpha =\pi n\,,n\in \mathbb {Z}$ ${\ displaystyle \ alpha = \ pi n \ ,, n \ in \ mathbb {Z}}$ .
Permanent intervals: $y>0$ ${\ displaystyle y> 0}$ at $\left(\pi n;{\frac {\pi }{2}}+\pi n\right)\,,n\in \mathbb {Z}$ ${\ displaystyle \ left (\ pi n; {\ frac {\ pi} {2}} + \ pi n \ right) \ ,, n \ in \ mathbb {Z}}$ and $y<0$ ${\ displaystyle y <0}$ at $\left(-{\frac {\pi }{2}}+\pi n;\pi n\right)\,,n\in \mathbb {Z}$ ${\ displaystyle \ left (- {\ frac {\ pi} {2}} + \ pi n; \ pi n \ right) \ ,, n \ in \ mathbb {Z}}$ .
The function is continuous and has a derivative for any value of the argument from the definition domain: $(\mathop {\operatorname {tg} } \,x)'={\frac {1}{\cos ^{2}x}}.$ ${\ displaystyle (\ mathop {\ operatorname {tg}} \, x) '= {\ frac {1} {\ cos ^ {2} x}}.}$
Function $y=\mathrm {tg} \ \alpha$ ${\ displaystyle y = \ mathrm {tg} \ \ alpha}$ increases with $\alpha \in \left(-{\frac {\pi }{2}}+\pi n;{\frac {\pi }{2}}+\pi n\right)\,,n\in \mathbb {Z}$ ${\ displaystyle \ alpha \ in \ left (- {\ frac {\ pi} {2}} + \ pi n; {\ frac {\ pi} {2}} + \ pi n \ right) \ ,, n \ in \ mathbb {Z}}$ .

Cotangent Function Properties

Cotangent

The domain of the function is the set of all real numbers: $D(y)=R,$ ${\ displaystyle D (y) = R,}$ except numbers $\alpha =\pi n,n\in \mathbb {Z} \,.$ ${\ displaystyle \ alpha = \ pi n, n \ in \ mathbb {Z} \ ,.}$
The set of values is the set of all real numbers: $E(y)=R.$ ${\ displaystyle E (y) = R.}$
Function $y=\mathop {\operatorname {ctg} } \left(\alpha \right)$ ${\ displaystyle y = \ mathop {\ operatorname {ctg}} \ left (\ alpha \ right)}$ is odd: $\mathop {\operatorname {ctg} } \left(-\alpha \right)=-\mathop {\operatorname {ctg} } \ \alpha \,.$ ${\ displaystyle \ mathop {\ operatorname {ctg}} \ left (- \ alpha \ right) = - \ mathop {\ operatorname {ctg}} \ \ alpha \ ,.}$
The function is periodic, the smallest positive period is $\pi$ ${\ displaystyle \ pi}$ : $\mathop {\operatorname {ctg} } \left(\alpha +\pi \right)=\mathop {\operatorname {ctg} } \left(\alpha \right).$ ${\ displaystyle \ mathop {\ operatorname {ctg}} \ left (\ alpha + \ pi \ right) = \ mathop {\ operatorname {ctg}} \ left (\ alpha \ right).}$
The function graph crosses the axis Ox at $\alpha ={\frac {\pi }{2}}+\pi n\,,n\in \mathbb {Z} \,.$ ${\ displaystyle \ alpha = {\ frac {\ pi} {2}} + \ pi n \ ,, n \ in \ mathbb {Z} \ ,.}$
Permanent intervals: $y>0$ ${\ displaystyle y> 0}$ at $\left(\pi n;{\frac {\pi }{2}}+\pi n\right)\,,n\in \mathbb {Z}$ ${\ displaystyle \ left (\ pi n; {\ frac {\ pi} {2}} + \ pi n \ right) \ ,, n \ in \ mathbb {Z}}$ and $y<0$ ${\ displaystyle y <0}$ at $\left({\frac {\pi }{2}}+\pi n;\pi \left(n+1\right)\right)\,,n\in \mathbb {Z} .$ ${\ displaystyle \ left ({\ frac {\ pi} {2}} + \ pi n; \ pi \ left (n + 1 \ right) \ right) \ ,, n \ in \ mathbb {Z}.}$
The function is continuous and has a derivative for any value of the argument from the definition domain: $(\mathop {\operatorname {ctg} } \,x)'=-{\frac {1}{\sin ^{2}x}}.$ ${\ displaystyle (\ mathop {\ operatorname {ctg}} \, x) '= - {\ frac {1} {\ sin ^ {2} x}}.}$
Function $y=\mathop {\operatorname {ctg} } \ \alpha$ ${\ displaystyle y = \ mathop {\ operatorname {ctg}} \ \ alpha}$ decreases with $\alpha \in \left(\pi n;\pi \left(n+1\right)\right)\,,n\in \mathbb {Z} .$ ${\ displaystyle \ alpha \ in \ left (\ pi n; \ pi \ left (n + 1 \ right) \ right) \ ,, n \ in \ mathbb {Z}.}$

Using trigonometry

Sextant - a navigation measuring tool used to measure the height of the star above the horizon in order to determine the geographical coordinates of the area in which the measurement is made.

There are many areas in which trigonometry and trigonometric functions are applied. For example, the triangulation method is used in astronomy to measure distances to nearby stars, in geography to measure distances between objects, and also in satellite navigation systems . Sine and cosine are fundamental to the theory of periodic functions , for example, in the description of sound and light waves.

Trigonometry or trigonometric functions are used in astronomy (especially for calculating the position of celestial objects when spherical trigonometry is required), in sea and air navigation, in the theory of music , in acoustics , in optics , in the analysis of financial markets , in electronics, in probability theory , statistics, in biology , in medical imaging (for example, computed tomography and ultrasound ), in pharmacies, in chemistry, in number theory (and therefore in cryptology ), in seismology , in meteorology , in oceanography , in many physical sciences, in land surveying and geodesy , in architecture , in phonetics , in economics , in electrical engineering , in mechanical engineering , in civil engineering, in computer graphics , in cartography , in crystallography , in game development and many other fields.

Standard Identities

Identities are equalities, valid for any values of the variables included in them.

\sin ^{2}A+\cos ^{2}A=1\ .

{\ displaystyle \ sin ^ {2} A + \ cos ^ {2} A = 1 \.}

\sec ^{2}A-{\mathop {\operatorname {tg} } }^{2}A=1\ .

{\ displaystyle \ sec ^ {2} A - {\ mathop {\ operatorname {tg}}} ^ {2} A = 1 \.}

\csc ^{2}A-{\mathop {\operatorname {ctg} } }^{2}A=1\ .

{\ displaystyle \ csc ^ {2} A - {\ mathop {\ operatorname {ctg}}} ^ {2} A = 1 \.}

Angle Sum Formulas

\sin(A\pm B)=\sin A\ \cos B\pm \cos A\ \sin B.

{\ displaystyle \ sin (A \ pm B) = \ sin A \ \ cos B \ pm \ cos A \ \ sin B.}

\cos(A\pm B)=\cos A\ \cos B\mp \sin A\ \sin B.

{\ displaystyle \ cos (A \ pm B) = \ cos A \ \ cos B \ mp \ sin A \ \ sin B.}

\mathop {\operatorname {tg} } (A\pm B)={\frac {\mathop {\operatorname {tg} } A\pm \mathop {\operatorname {tg} } B}{1\mp \mathop {\operatorname {tg} } A\ \mathop {\operatorname {tg} } B}}.

{\ displaystyle \ mathop {\ operatorname {tg}} (A \ pm B) = {\ frac {\ mathop {\ operatorname {tg}} A \ pm \ mathop {\ operatorname {tg}} B} {1 \ mp \ mathop {\ operatorname {tg}} A \ \ mathop {\ operatorname {tg}} B}}.}

\mathop {\operatorname {ctg} } (A\pm B)={\frac {\mathop {\operatorname {ctg} } A\ \mathop {\operatorname {ctg} } B\mp 1}{\mathop {\operatorname {ctg} } B\pm \mathop {\operatorname {ctg} } A}}.

{\ displaystyle \ mathop {\ operatorname {ctg}} (A \ pm B) = {\ frac {\ mathop {\ operatorname {ctg}} A \ \ mathop {\ operatorname {ctg}} B \ mp 1} {\ mathop {\ operatorname {ctg}} B \ pm \ mathop {\ operatorname {ctg}} A}}.}

General formulas

A triangle with sides a, b, c and correspondingly opposite corners A, B, C

In the following identities, A, B, and C are the angles of a triangle; a, b, c are the lengths of the sides of the triangle lying opposite the corresponding angles.

Sine Theorem

The sides of the triangle are proportional to the sines of the opposite angles . For an arbitrary triangle

{\frac {a}{\sin A}}={\frac {b}{\sin B}}={\frac {c}{\sin C}}=2R,

{\ displaystyle {\ frac {a} {\ sin A}} = {\ frac {b} {\ sin B}} = {\ frac {c} {\ sin C}} = 2R,}

Where $R$ ${\ displaystyle R}$ Is the radius of the circle circumscribed around the triangle.

R={\frac {abc}{\sqrt {(a+b+c)(a-b+c)(a+b-c)(b+c-a)}}}.

{\ displaystyle R = {\ frac {abc} {\ sqrt {(a + b + c) (a-b + c) (a + bc) (b + ca)}}}.}

Cosine Theorem

The square of the side of the triangle is the sum of the squares of the other two sides minus the double product of these sides by the cosine of the angle between them:

c^{2}=a^{2}+b^{2}-2ab\cos C,

{\ displaystyle c ^ {2} = a ^ {2} + b ^ {2} -2ab \ cos C,}

or:

\cos C={\frac {a^{2}+b^{2}-c^{2}}{2ab}}.

{\ displaystyle \ cos C = {\ frac {a ^ {2} + b ^ {2} -c ^ {2}} {2ab}}.}

Tangent Theorem

{\frac {a-b}{a+b}}={\frac {\mathop {\operatorname {tg} } \left[{\tfrac {1}{2}}(A-B)\right]}{\mathop {\operatorname {tg} } \left[{\tfrac {1}{2}}(A+B)\right]}}

{\ displaystyle {\ frac {ab} {a + b}} = {\ frac {\ mathop {\ operatorname {tg}} \ left [{\ tfrac {1} {2}} (AB) \ right]} { \ mathop {\ operatorname {tg}} \ left [{\ tfrac {1} {2}} (A + B) \ right]}}}

Euler Formula

Euler's formula states that for any real number $x$ ${\ displaystyle x}$ the following equality holds:

e^{ix}=\cos x+i\sin x,

{\ displaystyle e ^ {ix} = \ cos x + i \ sin x,}

Where $e$ ${\ displaystyle e}$ - the base of the natural logarithm , $i$ ${\ displaystyle i}$ - imaginary unit . The Euler formula provides a connection between mathematical analysis and trigonometry, and also allows you to interpret the sine and cosine functions as weighted sums of an exponential function:

\cos x=\mathrm {Re} \{e^{ix}\}={e^{ix}+e^{-ix} \over 2},

{\ displaystyle \ cos x = \ mathrm {Re} \ {e ^ {ix} \} = {e ^ {ix} + e ^ {- ix} \ over 2},}

\sin x=\mathrm {Im} \{e^{ix}\}={e^{ix}-e^{-ix} \over 2i}.

{\ displaystyle \ sin x = \ mathrm {Im} \ {e ^ {ix} \} = {e ^ {ix} -e ^ {- ix} \ over 2i}.}

The above equations can be obtained by adding or subtracting Euler's formulas:

e^{ix}=\cos x+i\sin x\;,

{\ displaystyle e ^ {ix} = \ cos x + i \ sin x \ ;,}

e^{-ix}=\cos(-x)+i\sin(-x)=\cos x-i\sin x\;.

{\ displaystyle e ^ {- ix} = \ cos (-x) + i \ sin (-x) = \ cos xi \ sin x \ ;.}

followed by a decision regarding sine or cosine.

Also, these formulas can serve as the definition of trigonometric functions of a complex variable. For example, performing the substitution x = iy , we obtain:

\cos iy={e^{-y}+e^{y} \over 2}=\operatorname {ch} y,

{\ displaystyle \ cos iy = {e ^ {- y} + e ^ {y} \ over 2} = \ operatorname {ch} y,}

\sin iy={e^{-y}-e^{y} \over 2i}=-{1 \over i}{e^{y}-e^{-y} \over 2}=i\operatorname {sh} y.

{\ displaystyle \ sin iy = {e ^ {- y} -e ^ {y} \ over 2i} = - {1 \ over i} {e ^ {y} -e ^ {- y} \ over 2} = i \ operatorname {sh} y.}

Complex exponents simplify trigonometric calculations, since they are easier to manipulate than sinusoidal components. One approach involves transforming sinusoids into corresponding exponential expressions. After simplification, the result of the expression remains real. The essence of another approach is to represent sinusoids as the material parts of a complex expression and to manipulate directly with a complex expression.

Solving Simple Trigonometric Equations

$\sin x=a.$ ${\ displaystyle \ sin x = a.}$

If a

|a|>1

{\ displaystyle | a |> 1}

- there are no material solutions.

If a

|a|\leqslant 1

{\ displaystyle | a | \ leqslant 1}

- the solution is a number of the form

x=(-1)^{n}\arcsin a+\pi n;\ n\in \mathbb {Z} .

{\ displaystyle x = (- 1) ^ {n} \ arcsin a + \ pi n; \ n \ in \ mathbb {Z}.}

$\cos x=a.$ ${\ displaystyle \ cos x = a.}$

If a

|a|>1

{\ displaystyle | a |> 1}

- there are no material solutions.

If a

|a|\leqslant 1

{\ displaystyle | a | \ leqslant 1}

- the solution is a number of the form

x=\pm \arccos a+2\pi n;\ n\in \mathbb {Z} .

{\ displaystyle x = \ pm \ arccos a + 2 \ pi n; \ n \ in \ mathbb {Z}.}

$\operatorname {tg} \,x=a.$ ${\ displaystyle \ operatorname {tg} \, x = a.}$

The solution is a number of the form

x=\operatorname {arctg} \,a+\pi n;\ n\in \mathbb {Z} .

{\ displaystyle x = \ operatorname {arctg} \, a + \ pi n; \ n \ in \ mathbb {Z}.}

$\operatorname {ctg} \,x=a.$ ${\ displaystyle \ operatorname {ctg} \, x = a.}$

The solution is a number of the form

x=\operatorname {arcctg} \,a+\pi n;\ n\in \mathbb {Z} .

{\ displaystyle x = \ operatorname {arcctg} \, a + \ pi n; \ n \ in \ mathbb {Z}.}

Spherical trigonometry

An important private section of trigonometry used in astronomy, geodesy, navigation, and other branches is spherical trigonometry, which considers the properties of the angles between large circles on a sphere and the arcs of these large circles. The geometry of the sphere is significantly different from Euclidean planimetry; so, the sum of the angles of a spherical triangle, generally speaking, differs from 180 °, a triangle can consist of three right angles. In spherical trigonometry, the lengths of the sides of a triangle (arcs of large circles of a sphere) are expressed by central angles corresponding to these arcs. Therefore, for example, the spherical sine theorem is expressed as

{\frac {\sin a}{\sin A}}={\frac {\sin b}{\sin B}}={\frac {\sin c}{\sin C}},

{\ displaystyle {\ frac {\ sin a} {\ sin A}} = {\ frac {\ sin b} {\ sin B}} = {\ frac {\ sin c} {\ sin C}},}

and there are two cosine theorems dual to each other.

Notes

↑ Soviet Encyclopedic Dictionary. M .: Soviet Encyclopedia, 1982.
↑ Boyer. Greek Trigonometry and Mensuration // Error: the |заглавие= parameter was not set in the template {{ publication }} . - 1991 .-- P. 162.

Literature

English

Boyer, Carl B. A History of Mathematics. - Second Edition. - John Wiley & Sons, Inc., 1991. - ISBN 0-471-54397-7 .
Christopher M. Linton (2004). From Eudoxus to Einstein: A History of Mathematical Astronomy. Cambridge University Press.
Weisstein, Eric W. Trigonometric Addition Formulas. Wolfram MathWorld. Weiner.

[1] Soviet Encyclopedic Dictionary. M .: Soviet Encyclopedia, 1982.

[2] Boyer. Greek Trigonometry and Mensuration // Error: the |заглавие= parameter was not set in the template {{ publication }} . - 1991 .-- P. 162.

Trigonometry

Content

History

Ancient Greece

Medieval India

Definition of trigonometric functions

Sine Function Properties

Cosine Function Properties

Tangent Function Properties

Cotangent Function Properties

Using trigonometry

Standard Identities

Angle Sum Formulas

General formulas

Sine Theorem

Cosine Theorem

Tangent Theorem

Euler Formula

Solving Simple Trigonometric Equations

Spherical trigonometry

See also

Notes

Literature

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