Reference data for tangent (tg x) and cotangent (ctg x). Geometric definition, properties, graphs, formulas. Table of tangents and cotangents, derivatives, integrals, series expansions. Expressions through complex variables. Connection with hyperbolic functions.
Geometric definition
|BD| - length of the arc of a circle with center at point A.
α is the angle expressed in radians.
Tangent ( tan α) is a trigonometric function depending on the angle α between the hypotenuse and the leg of a right triangle, equal to the ratio of the length of the opposite leg |BC| to the length of the adjacent leg |AB| .
Cotangent ( ctg α) is a trigonometric function depending on the angle α between the hypotenuse and the leg of a right triangle, equal to the ratio of the length of the adjacent leg |AB| to the length of the opposite leg |BC| .
Tangent
Where n- whole.
In Western literature, tangent is denoted as follows:
.
;
;
.
Graph of the tangent function, y = tan x
Cotangent
Where n- whole.
In Western literature, cotangent is denoted as follows:
.
The following notations are also accepted:
;
;
.
Graph of the cotangent function, y = ctg x
Properties of tangent and cotangent
Periodicity
Functions y = tg x and y = ctg x are periodic with period π.
Parity
The tangent and cotangent functions are odd.
Areas of definition and values, increasing, decreasing
The tangent and cotangent functions are continuous in their domain of definition (see proof of continuity). The main properties of tangent and cotangent are presented in the table ( n- whole).
| y = tg x | y = ctg x | |
| Scope and continuity | ||
| Range of values | -∞ < y < +∞ | -∞ < y < +∞ |
| Increasing | - | |
| Descending | - | |
| Extremes | - | - |
| Zeros, y = 0 | ||
| Intercept points with the ordinate axis, x = 0 | y = 0 | - |
Formulas
Expressions using sine and cosine
;
;
;
;
;
Formulas for tangent and cotangent from sum and difference
The remaining formulas are easy to obtain, for example
Product of tangents
Formula for the sum and difference of tangents
This table presents the values of tangents and cotangents for certain values of the argument. 
Expressions using complex numbers
Expressions through hyperbolic functions
;
;
Derivatives
; .
.
Derivative of the nth order with respect to the variable x of the function:
.
Deriving formulas for tangent > > > ; for cotangent > > >
Integrals
Series expansions
To obtain the expansion of the tangent in powers of x, you need to take several terms of the expansion in a power series for the functions sin x And cos x and divide these polynomials by each other, . This produces the following formulas.
At .
at .
Where Bn- Bernoulli numbers. They are determined either from the recurrence relation:
;
;
Where .
Or according to Laplace's formula: 
Inverse functions
The inverse functions of tangent and cotangent are arctangent and arccotangent, respectively.
Arctangent, arctg
, Where n- whole.
Arccotangent, arcctg
, Where n- whole.
Used literature:
I.N. Bronstein, K.A. Semendyaev, Handbook of mathematics for engineers and college students, “Lan”, 2009.
G. Korn, Handbook of Mathematics for Scientists and Engineers, 2012.
Sine is one of the basic trigonometric functions, the use of which is not limited to geometry alone. Tables for calculating trigonometric functions, like engineering calculators, are not always at hand, and calculating the sine is sometimes needed to solve various problems. In general, calculating the sine will help consolidate drawing skills and knowledge of trigonometric identities.
Games with ruler and pencil
A simple task: how to find the sine of an angle drawn on paper? To solve, you will need a regular ruler, a triangle (or compass) and a pencil. The simplest way to calculate the sine of an angle is by dividing the far leg of a triangle with a right angle by the long side - the hypotenuse. Thus, you first need to complete the acute angle to the shape of a right triangle by drawing a line perpendicular to one of the rays at an arbitrary distance from the vertex of the angle. We will need to maintain an angle of exactly 90°, for which we need a clerical triangle.
Using a compass is a little more accurate, but will take more time. On one of the rays you need to mark 2 points at a certain distance, set a radius on the compass approximately equal to the distance between the points, and draw semicircles with centers at these points until the intersections of these lines are obtained. By connecting the intersection points of our circles with each other, we get a strict perpendicular to the ray of our angle; all that remains is to extend the line until it intersects with another ray.
In the resulting triangle, you need to use a ruler to measure the side opposite the corner and the long side on one of the rays. The ratio of the first dimension to the second will be the desired value of the sine of the acute angle.
Find the sine for an angle greater than 90°
For an obtuse angle the task is not much more difficult. We need to draw a ray from the vertex in the opposite direction using a ruler to form a straight line with one of the rays of the angle we are interested in. The resulting acute angle should be treated as described above; the sines of adjacent angles that together form a reverse angle of 180° are equal.
Calculating sine using other trigonometric functions
Also, calculating the sine is possible if the values of other trigonometric functions of the angle or at least the lengths of the sides of the triangle are known. Trigonometric identities will help us with this. Let's look at common examples.
How to find the sine with a known cosine of an angle? The first trigonometric identity, based on the Pythagorean theorem, states that the sum of the squares of the sine and cosine of the same angle is equal to one.
How to find the sine with a known tangent of an angle? The tangent is obtained by dividing the far side by the near side or dividing the sine by the cosine. Thus, the sine will be the product of the cosine and the tangent, and the square of the sine will be the square of this product. We replace the squared cosine with the difference between unity and the square sine according to the first trigonometric identity and, through simple manipulations, we reduce the equation to the calculation of the square sine through the tangent; accordingly, to calculate the sine, you will have to extract the root of the result obtained.
How to find the sine with a known cotangent of an angle? The value of the cotangent can be calculated by dividing the length of the leg closest to the angle by the length of the far one, as well as dividing the cosine by the sine, that is, the cotangent is a function inverse to the tangent relative to the number 1. To calculate the sine, you can calculate the tangent using the formula tg α = 1 / ctg α and use the formula in the second option. You can also derive a direct formula by analogy with tangent, which will look like this.
How to find the sine of three sides of a triangle
There is a formula for finding the length of the unknown side of any triangle, not just a right triangle, from two known sides using the trigonometric function of the cosine of the opposite angle. She looks like this.
One of the areas of mathematics that students struggle with the most is trigonometry. It is not surprising: in order to freely master this area of knowledge, you need spatial thinking, the ability to find sines, cosines, tangents, cotangents using formulas, simplify expressions, and be able to use the number pi in calculations. In addition, you need to be able to use trigonometry when proving theorems, and this requires either a developed mathematical memory or the ability to derive complex logical chains.
Origins of trigonometry
Getting acquainted with this science should begin with the definition of sine, cosine and tangent of an angle, but first you need to understand what trigonometry does in general.
Historically, the main object of study in this branch of mathematical science was right triangles. The presence of an angle of 90 degrees makes it possible to carry out various operations that allow one to determine the values of all parameters of the figure in question using two sides and one angle or two angles and one side. In the past, people noticed this pattern and began to actively use it in the construction of buildings, navigation, astronomy and even in art.
Initial stage
Initially, people talked about the relationship between angles and sides solely using the example of right triangles. Then special formulas were discovered that made it possible to expand the boundaries of use in everyday life of this branch of mathematics.
The study of trigonometry in school today begins with right triangles, after which students use the acquired knowledge in physics and solving abstract trigonometric equations, which begin in high school.
Spherical trigonometry
Later, when science reached the next level of development, formulas with sine, cosine, tangent, and cotangent began to be used in spherical geometry, where different rules apply, and the sum of the angles in a triangle is always more than 180 degrees. This section is not studied in school, but it is necessary to know about its existence at least because the earth’s surface, and the surface of any other planet, is convex, which means that any surface marking will be “arc-shaped” in three-dimensional space.
Take the globe and the thread. Attach the thread to any two points on the globe so that it is taut. Please note - it has taken on the shape of an arc. Spherical geometry deals with such forms, which is used in geodesy, astronomy and other theoretical and applied fields.
Right triangle
Having learned a little about the ways of using trigonometry, let's return to basic trigonometry in order to further understand what sine, cosine, tangent are, what calculations can be performed with their help and what formulas to use.
The first step is to understand the concepts related to a right triangle. First, the hypotenuse is the side opposite the 90 degree angle. It is the longest. We remember that according to the Pythagorean theorem, its numerical value is equal to the root of the sum of the squares of the other two sides.
For example, if the two sides are 3 and 4 centimeters respectively, the length of the hypotenuse will be 5 centimeters. By the way, the ancient Egyptians knew about this about four and a half thousand years ago.
The two remaining sides, which form a right angle, are called legs. In addition, we must remember that the sum of the angles in a triangle in a rectangular coordinate system is equal to 180 degrees.
Definition
Finally, with a firm understanding of the geometric basis, one can turn to the definition of sine, cosine and tangent of an angle.
The sine of an angle is the ratio of the opposite leg (i.e., the side opposite the desired angle) to the hypotenuse. The cosine of an angle is the ratio of the adjacent side to the hypotenuse.
Remember that neither sine nor cosine can be greater than one! Why? Because the hypotenuse is by default the longest. No matter how long the leg is, it will be shorter than the hypotenuse, which means their ratio will always be less than one. Thus, if in your answer to a problem you get a sine or cosine with a value greater than 1, look for an error in the calculations or reasoning. This answer is clearly incorrect.
Finally, the tangent of an angle is the ratio of the opposite side to the adjacent side. Dividing the sine by the cosine will give the same result. Look: according to the formula, we divide the length of the side by the hypotenuse, then divide by the length of the second side and multiply by the hypotenuse. Thus, we get the same relationship as in the definition of tangent.
Cotangent, accordingly, is the ratio of the side adjacent to the corner to the opposite side. We get the same result by dividing one by the tangent.
So, we have looked at the definitions of what sine, cosine, tangent and cotangent are, and we can move on to formulas.
The simplest formulas
In trigonometry you cannot do without formulas - how to find sine, cosine, tangent, cotangent without them? But this is exactly what is required when solving problems.
The first formula you need to know when starting to study trigonometry says that the sum of the squares of the sine and cosine of an angle is equal to one. This formula is a direct consequence of the Pythagorean theorem, but it saves time if you need to know the size of the angle rather than the side.
Many students cannot remember the second formula, which is also very popular when solving school problems: the sum of one and the square of the tangent of an angle is equal to one divided by the square of the cosine of the angle. Take a closer look: this is the same statement as in the first formula, only both sides of the identity were divided by the square of the cosine. It turns out that a simple mathematical operation makes the trigonometric formula completely unrecognizable. Remember: knowing what sine, cosine, tangent and cotangent are, transformation rules and several basic formulas, you can at any time derive the required more complex formulas on a sheet of paper.
Formulas for double angles and addition of arguments
Two more formulas that you need to learn are related to the values of sine and cosine for the sum and difference of angles. They are presented in the figure below. Please note that in the first case, sine and cosine are multiplied both times, and in the second, the pairwise product of sine and cosine is added.
There are also formulas associated with double angle arguments. They are completely derived from the previous ones - as a practice, try to get them yourself by taking the alpha angle equal to the beta angle.
Finally, note that double angle formulas can be rearranged to reduce the power of sine, cosine, tangent alpha.
Theorems
The two main theorems in basic trigonometry are the sine theorem and the cosine theorem. With the help of these theorems, you can easily understand how to find the sine, cosine and tangent, and therefore the area of the figure, and the size of each side, etc.
The sine theorem states that dividing the length of each side of a triangle by the opposite angle results in the same number. Moreover, this number will be equal to two radii of the circumscribed circle, that is, the circle containing all the points of a given triangle.
The cosine theorem generalizes the Pythagorean theorem, projecting it onto any triangles. It turns out that from the sum of the squares of the two sides, subtract their product multiplied by the double cosine of the adjacent angle - the resulting value will be equal to the square of the third side. Thus, the Pythagorean theorem turns out to be a special case of the cosine theorem.
Careless mistakes
Even knowing what sine, cosine and tangent are, it is easy to make a mistake due to absent-mindedness or an error in the simplest calculations. To avoid such mistakes, let's take a look at the most popular ones.
First, you should not convert fractions to decimals until you get the final result - you can leave the answer as a fraction unless otherwise stated in the conditions. Such a transformation cannot be called a mistake, but it should be remembered that at each stage of the problem new roots may appear, which, according to the author’s idea, should be reduced. In this case, you will waste your time on unnecessary mathematical operations. This is especially true for values such as the root of three or the root of two, because they are found in problems at every step. The same goes for rounding “ugly” numbers.
Further, note that the cosine theorem applies to any triangle, but not the Pythagorean theorem! If you mistakenly forget to subtract twice the product of the sides multiplied by the cosine of the angle between them, you will not only get a completely wrong result, but you will also demonstrate a complete lack of understanding of the subject. This is worse than a careless mistake.
Thirdly, do not confuse the values for angles of 30 and 60 degrees for sines, cosines, tangents, cotangents. Remember these values, because the sine of 30 degrees is equal to the cosine of 60, and vice versa. It is easy to confuse them, as a result of which you will inevitably get an erroneous result.
Application
Many students are in no hurry to start studying trigonometry because they do not understand its practical meaning. What is sine, cosine, tangent for an engineer or astronomer? These are concepts with which you can calculate the distance to distant stars, predict the fall of a meteorite, or send a research probe to another planet. Without them, it is impossible to build a building, design a car, calculate the load on a surface or the trajectory of an object. And these are just the most obvious examples! After all, trigonometry in one form or another is used everywhere, from music to medicine.
In conclusion
So you're sine, cosine, tangent. You can use them in calculations and successfully solve school problems.
The whole point of trigonometry comes down to the fact that using the known parameters of a triangle you need to calculate the unknowns. There are six parameters in total: the length of three sides and the size of three angles. The only difference in the tasks is that different input data are given.
You now know how to find sine, cosine, tangent based on the known lengths of the legs or hypotenuse. Since these terms mean nothing more than a ratio, and a ratio is a fraction, the main goal of a trigonometry problem is to find the roots of an ordinary equation or system of equations. And here regular school mathematics will help you.
The concepts of sine, cosine, tangent and cotangent are the main categories of trigonometry, a branch of mathematics, and are inextricably linked with the definition of angle. Mastery of this mathematical science requires memorization and understanding of formulas and theorems, as well as developed spatial thinking. This is why trigonometric calculations often cause difficulties for schoolchildren and students. To overcome them, you should become more familiar with trigonometric functions and formulas.
Concepts in trigonometry
To understand the basic concepts of trigonometry, you must first understand what a right triangle and an angle in a circle are, and why all basic trigonometric calculations are associated with them. A triangle in which one of the angles measures 90 degrees is rectangular. Historically, this figure was often used by people in architecture, navigation, art, and astronomy. Accordingly, by studying and analyzing the properties of this figure, people came to calculate the corresponding ratios of its parameters.
The main categories associated with right triangles are the hypotenuse and the legs. The hypotenuse is the side of a triangle opposite the right angle. The legs, respectively, are the remaining two sides. The sum of the angles of any triangle is always 180 degrees.
Spherical trigonometry is a section of trigonometry that is not studied in school, but in applied sciences such as astronomy and geodesy, scientists use it. The peculiarity of a triangle in spherical trigonometry is that it always has a sum of angles greater than 180 degrees.
Angles of a triangle
In a right triangle, the sine of an angle is the ratio of the leg opposite the desired angle to the hypotenuse of the triangle. Accordingly, cosine is the ratio of the adjacent leg and the hypotenuse. Both of these values always have a magnitude less than one, since the hypotenuse is always longer than the leg.
The tangent of an angle is a value equal to the ratio of the opposite side to the adjacent side of the desired angle, or sine to cosine. Cotangent, in turn, is the ratio of the adjacent side of the desired angle to the opposite side. The cotangent of an angle can also be obtained by dividing one by the tangent value.
Unit circle
A unit circle in geometry is a circle whose radius is equal to one. Such a circle is constructed in a Cartesian coordinate system, with the center of the circle coinciding with the origin point, and the initial position of the radius vector is determined along the positive direction of the X axis (abscissa axis). Each point on the circle has two coordinates: XX and YY, that is, the coordinates of the abscissa and ordinate. By selecting any point on the circle in the XX plane and dropping a perpendicular from it to the abscissa axis, we obtain a right triangle formed by the radius to the selected point (denoted by the letter C), the perpendicular drawn to the X axis (the intersection point is denoted by the letter G), and the segment the abscissa axis is between the origin of coordinates (the point is designated by the letter A) and the intersection point G. The resulting triangle ACG is a right triangle inscribed in a circle, where AG is the hypotenuse, and AC and GC are the legs. The angle between the radius of the circle AC and the segment of the abscissa axis with the designation AG is defined as α (alpha). So, cos α = AG/AC. Considering that AC is the radius of the unit circle, and it is equal to one, it turns out that cos α=AG. Likewise, sin α=CG.
In addition, knowing this data, you can determine the coordinate of point C on the circle, since cos α=AG, and sin α=CG, which means point C has the given coordinates (cos α;sin α). Knowing that the tangent is equal to the ratio of sine to cosine, we can determine that tan α = y/x, and cot α = x/y. By considering angles in a negative coordinate system, you can calculate that the sine and cosine values of some angles can be negative.
Calculations and basic formulas
Trigonometric function values
Having considered the essence of trigonometric functions through the unit circle, we can derive the values of these functions for some angles. The values are listed in the table below.
The simplest trigonometric identities
Equations in which there is an unknown value under the sign of the trigonometric function are called trigonometric. Identities with the value sin x = α, k - any integer:
- sin x = 0, x = πk.
- 2. sin x = 1, x = π/2 + 2πk.
- sin x = -1, x = -π/2 + 2πk.
- sin x = a, |a| > 1, no solutions.
- sin x = a, |a| ≦ 1, x = (-1)^k * arcsin α + πk.
Identities with the value cos x = a, where k is any integer:
- cos x = 0, x = π/2 + πk.
- cos x = 1, x = 2πk.
- cos x = -1, x = π + 2πk.
- cos x = a, |a| > 1, no solutions.
- cos x = a, |a| ≦ 1, x = ±arccos α + 2πk.
Identities with the value tg x = a, where k is any integer:
- tan x = 0, x = π/2 + πk.
- tan x = a, x = arctan α + πk.
Identities with the value ctg x = a, where k is any integer:
- cot x = 0, x = π/2 + πk.
- ctg x = a, x = arcctg α + πk.
Reduction formulas
This category of constant formulas denotes methods with which you can move from trigonometric functions of the form to functions of an argument, that is, reduce the sine, cosine, tangent and cotangent of an angle of any value to the corresponding indicators of the angle of the interval from 0 to 90 degrees for greater convenience of calculations.
Formulas for reducing functions for the sine of an angle look like this:
- sin(900 - α) = α;
- sin(900 + α) = cos α;
- sin(1800 - α) = sin α;
- sin(1800 + α) = -sin α;
- sin(2700 - α) = -cos α;
- sin(2700 + α) = -cos α;
- sin(3600 - α) = -sin α;
- sin(3600 + α) = sin α.
For cosine of angle:
- cos(900 - α) = sin α;
- cos(900 + α) = -sin α;
- cos(1800 - α) = -cos α;
- cos(1800 + α) = -cos α;
- cos(2700 - α) = -sin α;
- cos(2700 + α) = sin α;
- cos(3600 - α) = cos α;
- cos(3600 + α) = cos α.
The use of the above formulas is possible subject to two rules. First, if the angle can be represented as a value (π/2 ± a) or (3π/2 ± a), the value of the function changes:
- from sin to cos;
- from cos to sin;
- from tg to ctg;
- from ctg to tg.
The value of the function remains unchanged if the angle can be represented as (π ± a) or (2π ± a).
Secondly, the sign of the reduced function does not change: if it was initially positive, it remains so. Same with negative functions.
Addition formulas
These formulas express the values of sine, cosine, tangent and cotangent of the sum and difference of two rotation angles through their trigonometric functions. Typically the angles are denoted as α and β.
The formulas look like this:
- sin(α ± β) = sin α * cos β ± cos α * sin.
- cos(α ± β) = cos α * cos β ∓ sin α * sin.
- tan(α ± β) = (tg α ± tan β) / (1 ∓ tan α * tan β).
- ctg(α ± β) = (-1 ± ctg α * ctg β) / (ctg α ± ctg β).
These formulas are valid for any angles α and β.
Double and triple angle formulas
The double and triple angle trigonometric formulas are formulas that relate the functions of the angles 2α and 3α, respectively, to the trigonometric functions of the angle α. Derived from addition formulas:
- sin2α = 2sinα*cosα.
- cos2α = 1 - 2sin^2 α.
- tan2α = 2tgα / (1 - tan^2 α).
- sin3α = 3sinα - 4sin^3 α.
- cos3α = 4cos^3 α - 3cosα.
- tg3α = (3tgα - tg^3 α) / (1-tg^2 α).
Transition from sum to product
Considering that 2sinx*cosy = sin(x+y) + sin(x-y), simplifying this formula, we obtain the identity sinα + sinβ = 2sin(α + β)/2 * cos(α − β)/2. Similarly sinα - sinβ = 2sin(α - β)/2 * cos(α + β)/2; cosα + cosβ = 2cos(α + β)/2 * cos(α − β)/2; cosα — cosβ = 2sin(α + β)/2 * sin(α − β)/2; tanα + tanβ = sin(α + β) / cosα * cosβ; tgα - tgβ = sin(α - β) / cosα * cosβ; cosα + sinα = √2sin(π/4 ∓ α) = √2cos(π/4 ± α).
Transition from product to sum
These formulas follow from the identities of the transition of a sum to a product:
- sinα * sinβ = 1/2*;
- cosα * cosβ = 1/2*;
- sinα * cosβ = 1/2*.
Degree reduction formulas
In these identities, the square and cubic powers of sine and cosine can be expressed in terms of the sine and cosine of the first power of a multiple angle:
- sin^2 α = (1 - cos2α)/2;
- cos^2 α = (1 + cos2α)/2;
- sin^3 α = (3 * sinα - sin3α)/4;
- cos^3 α = (3 * cosα + cos3α)/4;
- sin^4 α = (3 - 4cos2α + cos4α)/8;
- cos^4 α = (3 + 4cos2α + cos4α)/8.
Universal substitution
Formulas for universal trigonometric substitution express trigonometric functions in terms of the tangent of a half angle.
- sin x = (2tgx/2) * (1 + tan^2 x/2), with x = π + 2πn;
- cos x = (1 - tan^2 x/2) / (1 + tan^2 x/2), where x = π + 2πn;
- tg x = (2tgx/2) / (1 - tg^2 x/2), where x = π + 2πn;
- cot x = (1 - tg^2 x/2) / (2tgx/2), with x = π + 2πn.
Special cases
Special cases of the simplest trigonometric equations are given below (k is any integer).
Quotients for sine:
| Sin x value | x value |
|---|---|
| 0 | πk |
| 1 | π/2 + 2πk |
| -1 | -π/2 + 2πk |
| 1/2 | π/6 + 2πk or 5π/6 + 2πk |
| -1/2 | -π/6 + 2πk or -5π/6 + 2πk |
| √2/2 | π/4 + 2πk or 3π/4 + 2πk |
| -√2/2 | -π/4 + 2πk or -3π/4 + 2πk |
| √3/2 | π/3 + 2πk or 2π/3 + 2πk |
| -√3/2 | -π/3 + 2πk or -2π/3 + 2πk |
Quotients for cosine:
| cos x value | x value |
|---|---|
| 0 | π/2 + 2πk |
| 1 | 2πk |
| -1 | 2 + 2πk |
| 1/2 | ±π/3 + 2πk |
| -1/2 | ±2π/3 + 2πk |
| √2/2 | ±π/4 + 2πk |
| -√2/2 | ±3π/4 + 2πk |
| √3/2 | ±π/6 + 2πk |
| -√3/2 | ±5π/6 + 2πk |
Quotients for tangent:
| tg x value | x value |
|---|---|
| 0 | πk |
| 1 | π/4 + πk |
| -1 | -π/4 + πk |
| √3/3 | π/6 + πk |
| -√3/3 | -π/6 + πk |
| √3 | π/3 + πk |
| -√3 | -π/3 + πk |
Quotients for cotangent:
| ctg x value | x value |
|---|---|
| 0 | π/2 + πk |
| 1 | π/4 + πk |
| -1 | -π/4 + πk |
| √3 | π/6 + πk |
| -√3 | -π/3 + πk |
| √3/3 | π/3 + πk |
| -√3/3 | -π/3 + πk |
Theorems
Theorem of sines
There are two versions of the theorem - simple and extended. Simple sine theorem: a/sin α = b/sin β = c/sin γ. In this case, a, b, c are the sides of the triangle, and α, β, γ are the opposite angles, respectively.
Extended sine theorem for an arbitrary triangle: a/sin α = b/sin β = c/sin γ = 2R. In this identity, R denotes the radius of the circle in which the given triangle is inscribed.
Cosine theorem
The identity is displayed as follows: a^2 = b^2 + c^2 - 2*b*c*cos α. In the formula, a, b, c are the sides of the triangle, and α is the angle opposite to side a.
Tangent theorem
The formula expresses the relationship between the tangents of two angles and the length of the sides opposite them. The sides are labeled a, b, c, and the corresponding opposite angles are α, β, γ. Formula of the tangent theorem: (a - b) / (a+b) = tan((α - β)/2) / tan((α + β)/2).
Cotangent theorem
Connects the radius of a circle inscribed in a triangle with the length of its sides. If a, b, c are the sides of the triangle, and A, B, C, respectively, are the angles opposite them, r is the radius of the inscribed circle, and p is the semi-perimeter of the triangle, the following identities are valid:
- cot A/2 = (p-a)/r;
- cot B/2 = (p-b)/r;
- cot C/2 = (p-c)/r.
Application
Trigonometry is not only a theoretical science associated with mathematical formulas. Its properties, theorems and rules are used in practice by various branches of human activity - astronomy, air and sea navigation, music theory, geodesy, chemistry, acoustics, optics, electronics, architecture, economics, mechanical engineering, measuring work, computer graphics, cartography, oceanography, and many others.
Sine, cosine, tangent and cotangent are the basic concepts of trigonometry, with the help of which one can mathematically express the relationships between the angles and lengths of the sides in a triangle, and find the required quantities through identities, theorems and rules.
Trigonometry is a branch of mathematical science that studies trigonometric functions and their use in geometry. The development of trigonometry began in ancient Greece. During the Middle Ages, scientists from the Middle East and India made important contributions to the development of this science.
This article is devoted to the basic concepts and definitions of trigonometry. It discusses the definitions of the basic trigonometric functions: sine, cosine, tangent and cotangent. Their meaning is explained and illustrated in the context of geometry.
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Initially, the definitions of trigonometric functions whose argument is an angle were expressed in terms of the ratio of the sides of a right triangle.
Definitions of trigonometric functions
The sine of an angle (sin α) is the ratio of the leg opposite this angle to the hypotenuse.
Cosine of the angle (cos α) - the ratio of the adjacent leg to the hypotenuse.
Angle tangent (t g α) - the ratio of the opposite side to the adjacent side.
Angle cotangent (c t g α) - the ratio of the adjacent side to the opposite side.
These definitions are given for the acute angle of a right triangle!
Let's give an illustration.
In a triangle ABC with right angle C, the sine of angle A is equal to the ratio of leg BC to hypotenuse AB.
The definitions of sine, cosine, tangent and cotangent allow you to calculate the values of these functions from the known lengths of the sides of the triangle.
Important to remember!
The range of values of sine and cosine is from -1 to 1. In other words, sine and cosine take values from -1 to 1. The range of values of tangent and cotangent is the entire number line, that is, these functions can take on any values.
The definitions given above apply to acute angles. In trigonometry, the concept of rotation angle is introduced, the value of which, unlike an acute angle, is not limited to 0 to 90 degrees. The rotation angle in degrees or radians is expressed by any real number from - ∞ to + ∞.
In this context, we can define sine, cosine, tangent and cotangent of an angle of arbitrary magnitude. Let us imagine a unit circle with its center at the origin of the Cartesian coordinate system.
The initial point A with coordinates (1, 0) rotates around the center of the unit circle through a certain angle α and goes to point A 1. The definition is given in terms of the coordinates of point A 1 (x, y).
Sine (sin) of the rotation angle
The sine of the rotation angle α is the ordinate of point A 1 (x, y). sin α = y
Cosine (cos) of the rotation angle
The cosine of the rotation angle α is the abscissa of point A 1 (x, y). cos α = x
Tangent (tg) of the rotation angle
The tangent of the angle of rotation α is the ratio of the ordinate of point A 1 (x, y) to its abscissa. t g α = y x
Cotangent (ctg) of the rotation angle
The cotangent of the rotation angle α is the ratio of the abscissa of point A 1 (x, y) to its ordinate. c t g α = x y
Sine and cosine are defined for any rotation angle. This is logical, because the abscissa and ordinate of a point after rotation can be determined at any angle. The situation is different with tangent and cotangent. The tangent is undefined when a point after rotation goes to a point with a zero abscissa (0, 1) and (0, - 1). In such cases, the expression for tangent t g α = y x simply does not make sense, since it contains division by zero. The situation is similar with cotangent. The difference is that the cotangent is not defined in cases where the ordinate of a point goes to zero.
Important to remember!
Sine and cosine are defined for any angles α.
Tangent is defined for all angles except α = 90° + 180° k, k ∈ Z (α = π 2 + π k, k ∈ Z)
Cotangent is defined for all angles except α = 180° k, k ∈ Z (α = π k, k ∈ Z)
When solving practical examples, do not say “sine of the angle of rotation α”. The words “angle of rotation” are simply omitted, implying that it is already clear from the context what is being discussed.
Numbers
What about the definition of sine, cosine, tangent and cotangent of a number, and not the angle of rotation?
Sine, cosine, tangent, cotangent of a number
Sine, cosine, tangent and cotangent of a number t is a number that is respectively equal to sine, cosine, tangent and cotangent in t radian.
For example, the sine of the number 10 π is equal to the sine of the rotation angle of 10 π rad.
There is another approach to determining the sine, cosine, tangent and cotangent of a number. Let's take a closer look at it.
Any real number t a point on the unit circle is associated with the center at the origin of the rectangular Cartesian coordinate system. Sine, cosine, tangent and cotangent are determined through the coordinates of this point.
The starting point on the circle is point A with coordinates (1, 0).
Positive number t
Negative number t corresponds to the point to which the starting point will go if it moves around the circle counterclockwise and passes the path t.
Now that the connection between a number and a point on a circle has been established, we move on to the definition of sine, cosine, tangent and cotangent.
Sine (sin) of t
Sine of a number t- ordinate of a point on the unit circle corresponding to the number t. sin t = y
Cosine (cos) of t
Cosine of a number t- abscissa of the point of the unit circle corresponding to the number t. cos t = x
Tangent (tg) of t
Tangent of a number t- the ratio of the ordinate to the abscissa of a point on the unit circle corresponding to the number t. t g t = y x = sin t cos t
The latest definitions are in accordance with and do not contradict the definition given at the beginning of this paragraph. Point on the circle corresponding to the number t, coincides with the point to which the starting point goes after turning by an angle t radian.
Trigonometric functions of angular and numeric argument
Each value of the angle α corresponds to a certain value of the sine and cosine of this angle. Just like all angles α other than α = 90 ° + 180 ° k, k ∈ Z (α = π 2 + π k, k ∈ Z) correspond to a certain tangent value. Cotangent, as stated above, is defined for all α except α = 180° k, k ∈ Z (α = π k, k ∈ Z).
We can say that sin α, cos α, t g α, c t g α are functions of the angle alpha, or functions of the angular argument.
Similarly, we can talk about sine, cosine, tangent and cotangent as functions of a numerical argument. Every real number t corresponds to a certain value of the sine or cosine of a number t. All numbers other than π 2 + π · k, k ∈ Z, correspond to a tangent value. Cotangent, similarly, is defined for all numbers except π · k, k ∈ Z.
Basic functions of trigonometry
Sine, cosine, tangent and cotangent are the basic trigonometric functions.
It is usually clear from the context which argument of the trigonometric function (angular argument or numeric argument) we are dealing with.
Let's return to the definitions given at the very beginning and the alpha angle, which lies in the range from 0 to 90 degrees. The trigonometric definitions of sine, cosine, tangent, and cotangent are entirely consistent with the geometric definitions given by the aspect ratios of a right triangle. Let's show it.
Let's take a unit circle with a center in a rectangular Cartesian coordinate system. Let's rotate the starting point A (1, 0) by an angle of up to 90 degrees and draw a perpendicular to the abscissa axis from the resulting point A 1 (x, y). In the resulting right triangle, the angle A 1 O H is equal to the angle of rotation α, the length of the leg O H is equal to the abscissa of the point A 1 (x, y). The length of the leg opposite the angle is equal to the ordinate of the point A 1 (x, y), and the length of the hypotenuse is equal to one, since it is the radius of the unit circle.
In accordance with the definition from geometry, the sine of angle α is equal to the ratio of the opposite side to the hypotenuse.
sin α = A 1 H O A 1 = y 1 = y
This means that determining the sine of an acute angle in a right triangle through the aspect ratio is equivalent to determining the sine of the rotation angle α, with alpha lying in the range from 0 to 90 degrees.
Similarly, the correspondence of definitions can be shown for cosine, tangent and cotangent.
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