In this article we will show how to give definitions of sine, cosine, tangent and cotangent of an angle and number in trigonometry. Here we will talk about notations, give examples of entries, and give graphic illustrations. In conclusion, let us draw a parallel between the definitions of sine, cosine, tangent and cotangent in trigonometry and geometry.
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Let's see how the idea of sine, cosine, tangent and cotangent is formed in a school mathematics course. In geometry lessons, the definition of sine, cosine, tangent and cotangent of an acute angle in a right triangle is given. And later trigonometry is studied, which talks about sine, cosine, tangent and cotangent of the angle of rotation and number. Let us present all these definitions, give examples and give the necessary comments.
From the geometry course we know the definitions of sine, cosine, tangent and cotangent of an acute angle in a right triangle. They are given as the ratio of the sides of a right triangle. Let us give their formulations.
Definition.
Sine of an acute angle in a right triangle is the ratio of the opposite side to the hypotenuse.
Definition.
Cosine of an acute angle in a right triangle is the ratio of the adjacent leg to the hypotenuse.
Definition.
Tangent of an acute angle in a right triangle– this is the ratio of the opposite side to the adjacent side.
Definition.
Cotangent of an acute angle in a right triangle- this is the ratio of the adjacent side to the opposite side.
The designations for sine, cosine, tangent and cotangent are also introduced there - sin, cos, tg and ctg, respectively.
For example, if ABC is a right triangle with right angle C, then the sine of the acute angle A is equal to the ratio of the opposite side BC to the hypotenuse AB, that is, sin∠A=BC/AB.
These definitions allow you to calculate the values of sine, cosine, tangent and cotangent of an acute angle from the known lengths of the sides of a right triangle, as well as from the known values of sine, cosine, tangent, cotangent and the length of one of the sides to find the lengths of the other sides. For example, if we knew that in a right triangle the leg AC is equal to 3 and the hypotenuse AB is equal to 7, then we could calculate the value of the cosine of the acute angle A by definition: cos∠A=AC/AB=3/7.
In trigonometry, they begin to look at the angle more broadly - they introduce the concept of angle of rotation. The magnitude of the rotation angle, unlike an acute angle, is not limited to 0 to 90 degrees; the rotation angle in degrees (and in radians) can be expressed by any real number from −∞ to +∞.
In this light, the definitions of sine, cosine, tangent and cotangent are given not of an acute angle, but of an angle of arbitrary size - the angle of rotation. They are given through the x and y coordinates of the point A 1, to which the so-called starting point A(1, 0) goes after its rotation by an angle α around the point O - the beginning of the rectangular Cartesian coordinate system and the center of the unit circle.
Definition.
Sine of rotation angleα is the ordinate of point A 1, that is, sinα=y.
Definition.
Cosine of the rotation angleα is called the abscissa of point A 1, that is, cosα=x.
Definition.
Tangent of rotation angleα is the ratio of the ordinate of point A 1 to its abscissa, that is, tanα=y/x.
Definition.
Cotangent of the rotation angleα is the ratio of the abscissa of point A 1 to its ordinate, that is, ctgα=x/y.
Sine and cosine are defined for any angle α, since we can always determine the abscissa and ordinate of the point, which is obtained by rotating the starting point by angle α. But tangent and cotangent are not defined for any angle. The tangent is not defined for angles α at which the starting point goes to a point with zero abscissa (0, 1) or (0, −1), and this occurs at angles 90°+180° k, k∈Z (π /2+π·k rad). Indeed, at such angles of rotation, the expression tgα=y/x does not make sense, since it contains division by zero. As for the cotangent, it is not defined for angles α at which the starting point goes to the point with the zero ordinate (1, 0) or (−1, 0), and this occurs for angles 180° k, k ∈Z (π·k rad).
So, sine and cosine are defined for any rotation angles, tangent is defined for all angles except 90°+180°k, k∈Z (π/2+πk rad), and cotangent is defined for all angles except 180° ·k , k∈Z (π·k rad).
The definitions include the designations already known to us sin, cos, tg and ctg, they are also used to designate sine, cosine, tangent and cotangent of the angle of rotation (sometimes you can find the designations tan and cotcorresponding to tangent and cotangent). So the sine of a rotation angle of 30 degrees can be written as sin30°, the entries tg(−24°17′) and ctgα correspond to the tangent of the rotation angle −24 degrees 17 minutes and the cotangent of the rotation angle α. Recall that when writing the radian measure of an angle, the designation “rad” is often omitted. For example, the cosine of a rotation angle of three pi rad is usually denoted cos3·π.
In conclusion of this point, it is worth noting that when talking about sine, cosine, tangent and cotangent of the angle of rotation, the phrase “angle of rotation” or the word “rotation” is often omitted. That is, instead of the phrase “sine of the rotation angle alpha,” the phrase “sine of the alpha angle” or, even shorter, “sine alpha” is usually used. The same applies to cosine, tangent, and cotangent.
We will also say that the definitions of sine, cosine, tangent and cotangent of an acute angle in a right triangle are consistent with the definitions just given for sine, cosine, tangent and cotangent of an angle of rotation ranging from 0 to 90 degrees. We will justify this.
Definition.
Sine, cosine, tangent and cotangent of a number t is a number equal to the sine, cosine, tangent and cotangent of the rotation angle in t radians, respectively.
For example, the cosine of the number 8·π by definition is a number equal to the cosine of the angle of 8·π rad. And the cosine of an angle of 8·π rad is equal to one, therefore, the cosine of the number 8·π is equal to 1.
There is another approach to determining the sine, cosine, tangent and cotangent of a number. It consists in the fact that each real number t is associated with a point on the unit circle with the center at the origin of the rectangular coordinate system, and sine, cosine, tangent and cotangent are determined through the coordinates of this point. Let's look at this in more detail.
Let us show how a correspondence is established between real numbers and points on a circle:
Now we move on to the definitions of sine, cosine, tangent and cotangent of the number t. Let us assume that the number t corresponds to a point on the circle A 1 (x, y) (for example, the number &pi/2; corresponds to the point A 1 (0, 1)).
Definition.
Sine of the number t is the ordinate of the point on the unit circle corresponding to the number t, that is, sint=y.
Definition.
Cosine of the number t is called the abscissa of the point of the unit circle corresponding to the number t, that is, cost=x.
Definition.
Tangent of the number t is the ratio of the ordinate to the abscissa of a point on the unit circle corresponding to the number t, that is, tgt=y/x. In another equivalent formulation, the tangent of a number t is the ratio of the sine of this number to the cosine, that is, tgt=sint/cost.
Definition.
Cotangent of the number t is the ratio of the abscissa to the ordinate of a point on the unit circle corresponding to the number t, that is, ctgt=x/y. Another formulation is this: the tangent of the number t is the ratio of the cosine of the number t to the sine of the number t: ctgt=cost/sint.
Here we note that the definitions just given are consistent with the definition given at the beginning of this paragraph. Indeed, the point on the unit circle corresponding to the number t coincides with the point obtained by rotating the starting point by an angle of t radians.
It is still worth clarifying this point. Let's say we have the entry sin3. How can we understand whether we are talking about the sine of the number 3 or the sine of the rotation angle of 3 radians? This is usually clear from the context, otherwise it is likely not of fundamental importance.
According to the definitions given in the previous paragraph, each angle of rotation α corresponds to a very specific value sinα, as well as the value cosα. In addition, all rotation angles other than 90°+180°k, k∈Z (π/2+πk rad) correspond to tgα values, and values other than 180°k, k∈Z (πk rad ) – values of ctgα . Therefore sinα, cosα, tanα and ctgα are functions of the angle α. In other words, these are functions of the angular argument.
We can speak similarly about the functions sine, cosine, tangent and cotangent of a numerical argument. Indeed, each real number t corresponds to a very specific value sint, as well as cost. In addition, all numbers other than π/2+π·k, k∈Z correspond to values tgt, and numbers π·k, k∈Z - values ctgt.
The functions sine, cosine, tangent and cotangent are called basic trigonometric functions.
It is usually clear from the context whether we are dealing with trigonometric functions of an angular argument or a numerical argument. Otherwise, we can think of the independent variable as both a measure of the angle (angular argument) and a numeric argument.
However, at school we mainly study numerical functions, that is, functions whose arguments, as well as their corresponding function values, are numbers. Therefore, if we are talking specifically about functions, then it is advisable to consider trigonometric functions as functions of numerical arguments.
If we consider the rotation angle α ranging from 0 to 90 degrees, then the definitions of sine, cosine, tangent and cotangent of the rotation angle in the context of trigonometry are fully consistent with the definitions of sine, cosine, tangent and cotangent of an acute angle in a right triangle, which are given in the geometry course. Let's justify this.
Let us depict the unit circle in the rectangular Cartesian coordinate system Oxy. Let's mark the starting point A(1, 0) . Let's rotate it by an angle α ranging from 0 to 90 degrees, we get point A 1 (x, y). Let us drop the perpendicular A 1 H from point A 1 to the Ox axis.
It is easy to see that in a right triangle, the angle A 1 OH is equal to the angle of rotation α, the length of the leg OH adjacent to this angle is equal to the abscissa of point A 1, that is, |OH|=x, the length of the leg A 1 H opposite to the angle is equal to the ordinate of point A 1, that is, |A 1 H|=y, and the length of the hypotenuse OA 1 is equal to one, since it is the radius of the unit circle. Then, by definition from geometry, the sine of an acute angle α in a right triangle A 1 OH is equal to the ratio of the opposite leg to the hypotenuse, that is, sinα=|A 1 H|/|OA 1 |=y/1=y. And by definition from trigonometry, the sine of the rotation angle α is equal to the ordinate of point A 1, that is, sinα=y. This shows that determining the sine of an acute angle in a right triangle is equivalent to determining the sine of the rotation angle α when α is from 0 to 90 degrees.
Similarly, it can be shown that the definitions of cosine, tangent and cotangent of an acute angle α are consistent with the definitions of cosine, tangent and cotangent of the rotation angle α.
References.
We will begin our study of trigonometry with the right triangle. Let's define what sine and cosine are, as well as tangent and cotangent of an acute angle. This is the basics of trigonometry.
Let us remind you that right angle is an angle equal to . In other words, half a turned angle.
Acute angle- smaller.
Obtuse angle- larger. In relation to such an angle, “obtuse” is not an insult, but a mathematical term :-)
Let's draw a right triangle. A right angle is usually denoted by . Please note that the side opposite the corner is indicated by the same letter, only small. So, the side lying opposite the angle is designated.
The angle is denoted by the corresponding Greek letter.
Hypotenuse of a right triangle is the side opposite the right angle.
Legs- sides lying opposite acute angles.
The leg lying opposite the angle is called opposite(relative to angle). The other leg, which lies on one of the sides of the angle, is called adjacent.
Sinus The acute angle in a right triangle is the ratio of the opposite side to the hypotenuse:
Cosine acute angle in a right triangle - the ratio of the adjacent leg to the hypotenuse:
Tangent acute angle in a right triangle - the ratio of the opposite side to the adjacent side:
Another (equivalent) definition: the tangent of an acute angle is the ratio of the sine of the angle to its cosine:
Cotangent acute angle in a right triangle - the ratio of the adjacent side to the opposite (or, which is the same, the ratio of cosine to sine):
Note the basic relationships for sine, cosine, tangent, and cotangent below. They will be useful to us when solving problems.
Let's prove some of them.
1. The sum of the angles of any triangle is equal to . Means, the sum of two acute angles of a right triangle is equal to .
2. On the one hand, as the ratio of the opposite side to the hypotenuse. On the other hand, since for the angle the leg will be adjacent.
We get that . In other words, .
3. Let's take the Pythagorean theorem: . Let's divide both parts by:
We got basic trigonometric identity:
Thus, knowing the sine of an angle, we can find its cosine, and vice versa.
4. Dividing both sides of the main trigonometric identity by , we obtain:
This means that if we are given the tangent of an acute angle, then we can immediately find its cosine.
Likewise,
Okay, we have given definitions and written down formulas. But why do we still need sine, cosine, tangent and cotangent?
We know that the sum of the angles of any triangle is equal to.
We know the relationship between parties right triangle. This is the Pythagorean theorem: .
It turns out that knowing two angles in a triangle, you can find the third. Knowing the two sides of a right triangle, you can find the third. This means that the angles have their own ratio, and the sides have their own. But what should you do if in a right triangle you know one angle (except the right angle) and one side, but you need to find the other sides?
This is what people in the past encountered when making maps of the area and the starry sky. After all, it is not always possible to directly measure all sides of a triangle.
Sine, cosine and tangent - they are also called trigonometric angle functions- give relationships between parties And corners triangle. Knowing the angle, you can find all its trigonometric functions using special tables. And knowing the sines, cosines and tangents of the angles of a triangle and one of its sides, you can find the rest.
We will also draw a table of the values of sine, cosine, tangent and cotangent for “good” angles from to.
Please note the two red dashes in the table. At appropriate angle values, tangent and cotangent do not exist.
Let's look at several trigonometry problems from the FIPI Task Bank.
1. In a triangle, the angle is , . Find .
The problem is solved in four seconds.
Since , we have: .
2. In a triangle, the angle is , , . Find . , is equal half of the hypotenuse.
A triangle with angles , and is isosceles. In it, the hypotenuse is times larger than the leg.
First, consider a circle with radius 1 and center at (0;0). For any αЄR, the radius 0A can be drawn so that the radian measure of the angle between 0A and the 0x axis is equal to α. The counterclockwise direction is considered positive. Let the end of radius A have coordinates (a,b).
Definition: The number b, equal to the ordinate of the unit radius constructed in the described way, is denoted by sinα and is called the sine of angle α.
Example: sin 3π cos3π/2 = 0 0 = 0
Definition: The number a, equal to the abscissa of the end of the unit radius constructed in the described way, is denoted by cosα and is called the cosine of the angle α.
Example: cos0 cos3π + cos3.5π = 1 (-1) + 0 = 2
These examples use the definition of the sine and cosine of an angle in terms of the coordinates of the end of the unit radius and the unit circle. For a more visual representation, you need to draw a unit circle and plot the corresponding points on it, and then count their abscissas to calculate the cosine and ordinates to calculate the sine.
Definition: The function tgx=sinx/cosx for x≠π/2+πk, kЄZ, is called the cotangent of the angle x. The domain of definition of the function tgx is all real numbers, except x=π/2+πn, nЄZ.
Example: tg0 tgπ = 0 0 = 0
This example is similar to the previous one. To calculate the tangent of an angle, you need to divide the ordinate of a point by its abscissa.
Definition: The function ctgx=cosx/sinx for x≠πk, kЄZ is called the cotangent of the angle x. The domain of definition of the function ctgx = is all real numbers except points x=πk, kЄZ.
To make it clearer what cosine, sine, tangent and cotangent are. Let's consider an example on a regular right triangle with angle y and sides a,b,c. Hypotenuse c, legs a and b respectively. The angle between the hypotenuse c and the leg b y.
Definition: The sine of the angle y is the ratio of the opposite side to the hypotenuse: siny = a/c
Definition: The cosine of the angle y is the ratio of the adjacent leg to the hypotenuse: cosy= in/c
Definition: The tangent of the angle y is the ratio of the opposite side to the adjacent side: tgy = a/b
Definition: The cotangent of the angle y is the ratio of the adjacent side to the opposite side: ctgy= in/a
Sine, cosine, tangent and cotangent are also called trigonometric functions. Each angle has its own sine and cosine. And almost everyone has their own tangent and cotangent.
It is believed that if we are given an angle, then its sine, cosine, tangent and cotangent are known to us! And vice versa. Given a sine, or any other trigonometric function, respectively, we know the angle. Even special tables have been created where trigonometric functions are written for each angle.
Sine and cosine originally arose from the need to calculate quantities in right triangles. It was noticed that if the degree measure of the angles in a right triangle is not changed, then the aspect ratio, no matter how much these sides change in length, always remains the same.
This is how the concepts of sine and cosine were introduced. The sine of an acute angle in a right triangle is the ratio of the opposite side to the hypotenuse, and the cosine is the ratio of the side adjacent to the hypotenuse.
But cosines and sines can be used for more than just right triangles. To find the value of an obtuse or acute angle or side of any triangle, it is enough to apply the theorem of cosines and sines.
The cosine theorem is quite simple: “The square of a side of a triangle is equal to the sum of the squares of the other two sides minus twice the product of those sides and the cosine of the angle between them.”
There are two interpretations of the sine theorem: small and extended. According to the minor: “In a triangle, the angles are proportional to the opposite sides.” This theorem is often expanded due to the property of the circumscribed circle of a triangle: “In a triangle, the angles are proportional to the opposite sides, and their ratio is equal to the diameter of the circumscribed circle.”
The derivative is a mathematical tool that shows how quickly a function changes relative to a change in its argument. Derivatives are used in geometry, and in a number of technical disciplines.
When solving problems, you need to know the tabular values of the derivatives of trigonometric functions: sine and cosine. The derivative of a sine is a cosine, and a cosine is a sine, but with a minus sign.
Sines and cosines are especially often used in solving right triangles and problems related to them.
The convenience of sines and cosines is also reflected in technology. Angles and sides were easy to evaluate using the cosine and sine theorems, breaking down complex shapes and objects into “simple” triangles. Engineers who often deal with calculations of aspect ratios and degree measures spent a lot of time and effort calculating the cosines and sines of non-tabular angles.
Then Bradis tables came to the rescue, containing thousands of values of sines, cosines, tangents and cotangents of different angles. In Soviet times, some teachers forced their students to memorize pages of Bradis tables.
Radian is the angular value of an arc whose length is equal to the radius or 57.295779513° degrees.
Degree (in geometry) - 1/360th part of a circle or 1/90th part of a right angle.
π = 3.141592653589793238462… (approximate value of Pi).
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.
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.
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.
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.
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.
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:
Identities with the value cos x = a, where k is any integer:
Identities with the value tg x = a, where k is any integer:
Identities with the value ctg x = a, where k is any integer:
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:
For cosine of angle:
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:
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.
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:
These formulas are valid for any angles α and β.
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:
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).
These formulas follow from the identities of the transition of a sum to a product:
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:
Formulas for universal trigonometric substitution express trigonometric functions in terms of the tangent of a half angle.
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 |
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.
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.
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).
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:
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.
