To solve some problems, a table of trigonometric identities will be useful, which will make it much easier to transform functions:

The simplest trigonometric identities

The quotient of dividing the sine of an angle alpha by the cosine of the same angle is equal to the tangent of this angle (Formula 1). See also the proof of the correctness of the transformation of the simplest trigonometric identities.
The quotient of dividing the cosine of an angle alpha by the sine of the same angle is equal to the cotangent of the same angle (Formula 2)
The secant of an angle is equal to one divided by the cosine of the same angle (Formula 3)
The sum of the squares of the sine and cosine of the same angle is equal to one (Formula 4). see also the proof of the sum of the squares of cosine and sine.
The sum of one and the tangent of an angle is equal to the ratio of one to the square of the cosine of this angle (Formula 5)
One plus the cotangent of an angle is equal to the quotient of one divided by the sine square of this angle (Formula 6)
The product of tangent and cotangent of the same angle is equal to one (Formula 7).

Converting negative angles of trigonometric functions (even and odd)

To get rid of the negative value degree measure angle when calculating sine, cosine or tangent, you can use the following trigonometric transformations (identities) based on the principles of even or odd trigonometric functions.

As seen, cosine and the secant is even function , sine, tangent and cotangent are odd functions.

The sine of a negative angle is equal to negative value sine of the same positive angle (minus sine alpha).
The cosine minus alpha will give the same value as the cosine of the alpha angle.
Tangent minus alpha is equal to minus tangent alpha.

Formulas for reducing double angles (sine, cosine, tangent and cotangent of double angles)

If you need to divide an angle in half, or vice versa, move from a double angle to a single angle, you can use the following trigonometric identities:

Double Angle Conversion (sine of a double angle, cosine of a double angle and tangent of a double angle) in single occurs by following rules:

Sine of double angle equal to twice the product of the sine and the cosine of a single angle

Cosine of double angle equal to the difference between the square of the cosine of a single angle and the square of the sine of this angle

Cosine of double angle equal to twice the square of the cosine of a single angle minus one

Cosine of double angle equal to one minus double sine squared single angle

Tangent of double angle is equal to a fraction whose numerator is twice the tangent of a single angle, and the denominator is equal to one minus the tangent squared of a single angle.

Cotangent of double angle is equal to a fraction whose numerator is the square of the cotangent of a single angle minus one, and the denominator is equal to twice the cotangent of a single angle

Formulas for universal trigonometric substitution

The conversion formulas below can be useful when you need to divide the argument of a trigonometric function (sin α, cos α, tan α) by two and reduce the expression to the value of half an angle. From the value of α we obtain α/2.

These formulas are called formulas of universal trigonometric substitution. Their value lies in the fact that with their help a trigonometric expression is reduced to expressing the tangent of half an angle, regardless of what trigonometric functions (sin cos tan ctg) were originally in the expression. After this, the equation with the tangent of half an angle is much easier to solve.

Trigonometric identities for half-angle transformations

The following are the formulas for trigonometric conversion of half an angle to its whole value.
The value of the argument of the trigonometric function α/2 is reduced to the value of the argument of the trigonometric function α.

Trigonometric formulas for adding angles

cos (α - β) = cos α cos β + sin α sin β

sin (α + β) = sin α cos β + sin β cos α

sin (α - β) = sin α cos β - sin β cos α
cos (α + β) = cos α cos β - sin α sin β

Tangent and cotangent of the sum of angles alpha and beta can be converted using the following rules for converting trigonometric functions:

Tangent of the sum of angles is equal to a fraction whose numerator is the sum of the tangent of the first and tangent of the second angle, and the denominator is one minus the product of the tangent of the first angle and the tangent of the second angle.

Tangent of angle difference is equal to a fraction whose numerator is equal to the difference between the tangent of the angle being reduced and the tangent of the angle being subtracted, and the denominator is one plus the product of the tangents of these angles.

Cotangent of the sum of angles is equal to a fraction whose numerator is equal to the product of the cotangents of these angles plus one, and the denominator is equal to the difference between the cotangent of the second angle and the cotangent of the first angle.

Cotangent of angle difference is equal to a fraction, the numerator of which is the product of the cotangents of these angles minus one, and the denominator equal to the sum cotangents of these angles.

These trigonometric identities are convenient to use when you need to calculate, for example, the tangent of 105 degrees (tg 105). If you represent it as tg (45 + 60), then you can use the given identical transformations tangent of the sum of angles, then simply substitute the tabulated values ​​of tangent 45 and tangent 60 degrees.

Formulas for converting the sum or difference of trigonometric functions

Expressions representing a sum of the form sin α + sin β can be transformed using the following formulas:

Triple angle formulas - converting sin3α cos3α tan3α to sinα cosα tanα

Sometimes it is necessary to transform the triple value of an angle so that the argument of the trigonometric function becomes the angle α instead of 3α.
In this case, you can use the triple angle transformation formulas (identities):

Formulas for converting products of trigonometric functions

If there is a need to transform the product of sines of different angles, cosines of different angles, or even the product of sine and cosine, then you can use the following trigonometric identities:


In this case, the product of the sine, cosine or tangent functions of different angles will be converted into a sum or difference.

Formulas for reducing trigonometric functions

You need to use the reduction table as follows. In the line we select the function that interests us. In the column there is an angle. For example, the sine of the angle (α+90) at the intersection of the first row and the first column, we find out that sin (α+90) = cos α.

How to find sine?

Studying geometry helps develop thinking. This subject is necessarily included in school preparation. In everyday life, knowledge of this subject can be useful - for example, when planning an apartment.

From the history

The geometry course also includes trigonometry, which studies trigonometric functions. In trigonometry we study sines, cosines, tangents and cotangents of angles.

But on this moment Let's start with the simplest thing - sine. Let's take a closer look at the very first concept - the sine of an angle in geometry. What is sine and how to find it?

The concept of “sine angle” and sinusoids

The sine of an angle is the ratio of the values ​​of the opposite side and the hypotenuse of a right triangle. This is a direct trigonometric function, which is written as “sin (x)”, where (x) is the angle of the triangle.

On the graph, the sine of an angle is indicated by a sine wave with its own characteristics. A sine wave looks like a continuous wavy line that lies within certain limits on the coordinate plane. The function is odd, therefore it is symmetrical about 0 on the coordinate plane (it comes out from the origin of the coordinates).

The domain of definition of this function lies in the range from -1 to +1 on the Cartesian coordinate system. The period of the sine angle function is 2 Pi. This means that every 2 Pi the pattern repeats and the sine wave goes through a full cycle.

Sine wave equation

• sin x = a/c
• where a is the leg opposite to the angle of the triangle
• c - hypotenuse of a right triangle

Properties of the sine of an angle

1. sin(x) = - sin(x). This feature demonstrates that the function is symmetrical, and if the values ​​x and (-x) are plotted on the coordinate system in both directions, then the ordinates of these points will be opposite. They will be at an equal distance from each other.
2. Another feature of this function is that the graph of the function increases on the segment [- P/2 + 2 Pn]; [P/2 + 2Pn], where n is any integer. A decrease in the graph of the sine of the angle will be observed on the segment: [P/2 + 2Pn]; [3P/2 + 2Pn].
3. sin(x) > 0 when x is in the range (2Пn, П + 2Пn)
4. (x)< 0, когда х находится в диапазоне (-П+2Пn, 2Пn)

The values ​​of the sines of the angle are determined using special tables. Such tables have been created to facilitate the process of calculating complex formulas and equations. It is easy to use and contains not only the values ​​of the sin(x) function, but also the values ​​of other functions.

Moreover, a table of standard values ​​of these functions is included in the compulsory memory study, like a multiplication table. This is especially true for classes with a physical and mathematical bias. In the table you can see the values ​​of the main angles used in trigonometry: 0, 15, 30, 45, 60, 75, 90, 120, 135, 150, 180, 270 and 360 degrees.

There is also a table defining the values ​​of trigonometric functions of non-standard angles. Using different tables, you can easily calculate the sine, cosine, tangent and cotangent of some angles.

Equations are made with trigonometric functions. Solving these equations is easy if you know simple trigonometric identities and reductions of functions, for example, such as sin (P/2 + x) = cos (x) and others. A separate table has also been compiled for such reductions.

How to find the sine of an angle

When the task is to find the sine of an angle, and according to the condition we only have the cosine, tangent, or cotangent of the angle, we can easily calculate what we need using trigonometric identities.

• sin 2 x + cos 2 x = 1

From this equation, we can find both sine and cosine, depending on which value is unknown. We get a trigonometric equation with one unknown:

• sin 2 x = 1 - cos 2 x
• sin x = ± √ 1 - cos 2 x
• cot 2 x + 1 = 1 / sin 2 x

From this equation you can find the value of the sine, knowing the value of the cotangent of the angle. To simplify, replace sin 2 x = y and you have a simple equation. For example, the cotangent value is 1, then:

• 1 + 1 = 1/y
• 2 = 1/y
• 2у = 1
• y = 1/2

Now we perform the reverse replacement of the player:

• sin 2 x = ½
• sin x = 1 / √2

Since we took the cotangent value for the standard angle (45 0), the obtained values ​​can be checked in the table.

If you are given a tangent value and need to find the sine, another trigonometric identity will help:

• tg x * ctg x = 1

It follows that:

• cot x = 1 / tan x

In order to find the sine of a non-standard angle, for example, 240 0, you need to use angle reduction formulas. We know that π corresponds to 180 0. Thus, we express our equality using standard angles by expansion.

• 240 0 = 180 0 + 60 0

We need to find the following: sin (180 0 + 60 0). In trigonometry there are reduction formulas that in this case will come in handy. This is the formula:

• sin (π + x) = - sin (x)

Thus, the sine of an angle of 240 degrees is equal to:

• sin (180 0 + 60 0) = - sin (60 0) = - √3/2

In our case, x = 60, and P, respectively, 180 degrees. We found the value (-√3/2) from the table of values ​​of functions of standard angles.

In this way, non-standard angles can be expanded, for example: 210 = 180 + 30.

Basic trigonometry formulas are formulas that establish connections between basic trigonometric functions. Sine, cosine, tangent and cotangent are interconnected by many relationships. Below we present the main trigonometric formulas, and for convenience we will group them by purpose. Using these formulas you can solve almost any problem from a standard trigonometry course. Let us immediately note that below are only the formulas themselves, and not their conclusion, which will be discussed in separate articles.

Basic identities of trigonometry

Trigonometric identities provide a relationship between the sine, cosine, tangent and cotangent of one angle, allowing one function to be expressed in terms of another.

Trigonometric identities

sin 2 a + cos 2 a = 1 t g α = sin α cos α , c t g α = cos α sin α t g α c t g α = 1 t g 2 α + 1 = 1 cos 2 α , c t g 2 α + 1 = 1 sin 2 α

These identities follow directly from the definitions of the unit circle, sine (sin), cosine (cos), tangent (tg) and cotangent (ctg).

Reduction formulas

Reduction formulas allow you to move from working with arbitrary and arbitrarily large angles to working with angles ranging from 0 to 90 degrees.

Reduction formulas

sin α + 2 π z = sin α , cos α + 2 π z = cos α t g α + 2 π z = t g α , c t g α + 2 π z = c t g α sin - α + 2 π z = - sin α , cos - α + 2 π z = cos α t g - α + 2 π z = - t g α , c t g - α + 2 π z = - c t g α sin π 2 + α + 2 π z = cos α , cos π 2 + α + 2 π z = - sin α t g π 2 + α + 2 π z = - c t g α , c t g π 2 + α + 2 π z = - t g α sin π 2 - α + 2 π z = cos α , cos π 2 - α + 2 π z = sin α t g π 2 - α + 2 π z = c t g α , c t g π 2 - α + 2 π z = t g α sin π + α + 2 π z = - sin α , cos π + α + 2 π z = - cos α t g π + α + 2 π z = t g α , c t g π + α + 2 π z = c t g α sin π - α + 2 π z = sin α , cos π - α + 2 π z = - cos α t g π - α + 2 π z = - t g α , c t g π - α + 2 π z = - c t g α sin 3 π 2 + α + 2 π z = - cos α , cos 3 π 2 + α + 2 π z = sin α t g 3 π 2 + α + 2 π z = - c t g α , c t g 3 π 2 + α + 2 π z = - t g α sin 3 π 2 - α + 2 π z = - cos α , cos 3 π 2 - α + 2 π z = - sin α t g 3 π 2 - α + 2 π z = c t g α , c t g 3 π 2 - α + 2 π z = t g α

Reduction formulas are a consequence of the periodicity of trigonometric functions.

Trigonometric addition formulas

Addition formulas in trigonometry allow you to express the trigonometric function of the sum or difference of angles in terms of trigonometric functions of these angles.

Trigonometric addition formulas

sin α ± β = sin α · cos β ± cos α · sin β cos α + β = cos α · cos β - sin α · sin β cos α - β = cos α · cos β + sin α · sin β t g α ± β = t g α ± t g β 1 ± t g α t g β c t g α ± β = - 1 ± c t g α c t g β c t g α ± c t g β

Based on addition formulas, trigonometric formulas for multiple angles are derived.

Formulas for multiple angles: double, triple, etc.

Double and triple angle formulas

sin 2 α = 2 · sin α · cos α cos 2 α = cos 2 α - sin 2 α , cos 2 α = 1 - 2 sin 2 α , cos 2 α = 2 cos 2 α - 1 t g 2 α = 2 · t g α 1 - t g 2 α with t g 2 α = with t g 2 α - 1 2 · with t g α sin 3 α = 3 sin α · cos 2 α - sin 3 α , sin 3 α = 3 sin α - 4 sin 3 α cos 3 α = cos 3 α - 3 sin 2 α · cos α , cos 3 α = - 3 cos α + 4 cos 3 α t g 3 α = 3 t g α - t g 3 α 1 - 3 t g 2 α c t g 3 α = c t g 3 α - 3 c t g α 3 c t g 2 α - 1

Half angle formulas

Half-angle formulas in trigonometry are a consequence of double-angle formulas and express the relationship between the basic functions of a half-angle and the cosine of a whole angle.

Half angle formulas

sin 2 α 2 = 1 - cos α 2 cos 2 α 2 = 1 + cos α 2 t g 2 α 2 = 1 - cos α 1 + cos α c t g 2 α 2 = 1 + cos α 1 - cos α

Degree reduction formulas

Degree reduction formulas

sin 2 α = 1 - cos 2 α 2 cos 2 α = 1 + cos 2 α 2 sin 3 α = 3 sin α - sin 3 α 4 cos 3 α = 3 cos α + cos 3 α 4 sin 4 α = 3 - 4 cos 2 α + cos 4 α 8 cos 4 α = 3 + 4 cos 2 α + cos 4 α 8

It is often inconvenient to work with cumbersome powers when making calculations. Degree reduction formulas allow you to reduce the degree of a trigonometric function from arbitrarily large to the first. Here is their general view:

General view of the degree reduction formulas

for even n

sin n α = C n 2 n 2 n + 1 2 n - 1 ∑ k = 0 n 2 - 1 (- 1) n 2 - k · C k n · cos ((n - 2 k) α) cos n α = C n 2 n 2 n + 1 2 n - 1 ∑ k = 0 n 2 - 1 C k n cos ((n - 2 k) α)

for odd n

sin n α = 1 2 n - 1 ∑ k = 0 n - 1 2 (- 1) n - 1 2 - k C k n sin ((n - 2 k) α) cos n α = 1 2 n - 1 ∑ k = 0 n - 1 2 C k n cos ((n - 2 k) α)

Sum and difference of trigonometric functions

The difference and sum of trigonometric functions can be represented as a product. Factoring differences of sines and cosines is very convenient to use when solving trigonometric equations and simplifying expressions.

Sum and difference of trigonometric functions

sin α + sin β = 2 sin α + β 2 cos α - β 2 sin α - sin β = 2 sin α - β 2 cos α + β 2 cos α + cos β = 2 cos α + β 2 cos α - β 2 cos α - cos β = - 2 sin α + β 2 sin α - β 2 , cos α - cos β = 2 sin α + β 2 sin β - α 2

Product of trigonometric functions

If the formulas for the sum and difference of functions allow one to go to their product, then the formulas for the product of trigonometric functions carry out the reverse transition - from the product to the sum. Formulas for the product of sines, cosines and sine by cosine are considered.

Formulas for the product of trigonometric functions

sin α · sin β = 1 2 · (cos (α - β) - cos (α + β)) cos α · cos β = 1 2 · (cos (α - β) + cos (α + β)) sin α cos β = 1 2 (sin (α - β) + sin (α + β))

Universal trigonometric substitution

All basic trigonometric functions - sine, cosine, tangent and cotangent - can be expressed in terms of the tangent of a half angle.

Universal trigonometric substitution

sin α = 2 t g α 2 1 + t g 2 α 2 cos α = 1 - t g 2 α 2 1 + t g 2 α 2 t g α = 2 t g α 2 1 - t g 2 α 2 c t g α = 1 - t g 2 α 2 2 t g α 2

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Table of values ​​of trigonometric functions

Note. This table of trigonometric function values ​​uses the √ sign to indicate square root. To indicate a fraction, use the symbol "/".

see also useful materials:

For determining the value of a trigonometric function, find it at the intersection of the line indicating the trigonometric function. For example, sine 30 degrees - we look for the column with the heading sin (sine) and find the intersection of this table column with the row “30 degrees”, at their intersection we read the result - one half. Similarly we find cosine 60 degrees, sine 60 degrees (once again, at the intersection of the sin column and the 60 degree line we find the value sin 60 = √3/2), etc. The values ​​of sines, cosines and tangents of other “popular” angles are found in the same way.

Sine pi, cosine pi, tangent pi and other angles in radians

The table below of cosines, sines and tangents is also suitable for finding the value of trigonometric functions whose argument is given in radians. To do this, use the second column of angle values. Thanks to this, you can convert the value of popular angles from degrees to radians. For example, let's find the angle of 60 degrees in the first line and read its value in radians under it. 60 degrees is equal to π/3 radians.

The number pi unambiguously expresses the dependence of the circumference on the degree measure of the angle. Thus, pi radians are equal to 180 degrees.

Any number expressed in terms of pi (radians) can be easily converted to degrees by replacing pi (π) with 180.

Examples:
1. Sine pi.
sin π = sin 180 = 0
thus, the sine of pi is the same as the sine of 180 degrees and it is equal to zero.

2. Cosine pi.
cos π = cos 180 = -1
thus, the cosine of pi is the same as the cosine of 180 degrees and it is equal to minus one.

3. Tangent pi
tg π = tg 180 = 0
thus, tangent pi is the same as tangent 180 degrees and it is equal to zero.

Table of sine, cosine, tangent values ​​for angles 0 - 360 degrees (common values)

angle α value (degrees)	angle α value in radians (via pi)	sin (sinus)	cos (cosine)	tg (tangent)	ctg (cotangent)	sec (secant)	cosec (cosecant)
0	0	0	1	0	-	1	-
15	π/12			2 - √3	2 + √3
30	π/6	1/2	√3/2	1/√3	√3	2/√3	2
45	π/4	√2/2	√2/2	1	1	√2	√2
60	π/3	√3/2	1/2	√3	1/√3	2	2/√3
75	5π/12			2 + √3	2 - √3
90	π/2	1	0	-	0	-	1
105	7π/12		-	- 2 - √3	√3 - 2
120	2π/3	√3/2	-1/2	-√3	-√3/3
135	3π/4	√2/2	-√2/2	-1	-1	-√2	√2
150	5π/6	1/2	-√3/2	-√3/3	-√3
180	π	0	-1	0	-	-1	-
210	7π/6	-1/2	-√3/2	√3/3	√3
240	4π/3	-√3/2	-1/2	√3	√3/3
270	3π/2	-1	0	-	0	-	-1
360	2π	0	1	0	-	1	-

If in the table of values ​​of trigonometric functions a dash is indicated instead of the function value (tangent (tg) 90 degrees, cotangent (ctg) 180 degrees), then for a given value of the degree measure of the angle the function does not have a specific value. If there is no dash, the cell is empty, which means we have not yet entered the required value. We are interested in what queries users come to us for and supplement the table with new values, despite the fact that current data on the values ​​of cosines, sines and tangents of the most common angle values ​​is quite sufficient to solve most problems.

Table of values ​​of trigonometric functions sin, cos, tg for the most popular angles
0, 15, 30, 45, 60, 90 ... 360 degrees
(numeric values ​​“as per Bradis tables”)

angle α value (degrees)	angle α value in radians	sin (sine)	cos (cosine)	tg (tangent)	ctg (cotangent)
0	0
15		0,2588	0,9659	0,2679
30		0,5000		0,5774
45		0,7071
		0,7660
60		0,8660	0,5000	1,7321
	7π/18

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. Hypotenuse - the side of a triangle opposite right angle. The legs, respectively, are the other two sides. The sum of the angles of any triangles 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 between the origin (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:

1. sin x = 0, x = πk.
2. 2. sin x = 1, x = π/2 + 2πk.
3. sin x = -1, x = -π/2 + 2πk.
4. sin x = a, |a| > 1, no solutions.
5. sin x = a, |a| ≦ 1, x = (-1)^k * arcsin α + πk.

Identities with the value cos x = a, where k is any integer:

1. cos x = 0, x = π/2 + πk.
2. cos x = 1, x = 2πk.
3. cos x = -1, x = π + 2πk.
4. cos x = a, |a| > 1, no solutions.
5. cos x = a, |a| ≦ 1, x = ±arccos α + 2πk.

Identities with the value tg x = a, where k is any integer:

1. tan x = 0, x = π/2 + πk.
2. tan x = a, x = arctan α + πk.

Identities with the value ctg x = a, where k is any integer:

1. cot x = 0, x = π/2 + πk.
2. ctg x = a, x = arcctg α + πk.

Reduction formulas

This category constant formulas denotes methods by 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:

1. sin(α ± β) = sin α * cos β ± cos α * sin.
2. cos(α ± β) = cos α * cos β ∓ sin α * sin.
3. tan(α ± β) = (tg α ± tan β) / (1 ∓ tan α * tan β).
4. 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:

1. sin2α = 2sinα*cosα.
2. cos2α = 1 - 2sin^2 α.
3. tan2α = 2tgα / (1 - tan^2 α).
4. sin3α = 3sinα - 4sin^3 α.
5. cos3α = 4cos^3 α - 3cosα.
6. 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.